# Spectral Functions for the Vector-Valued Fourier Transform

Authors: Vadim Mogilevskii
Journal: Journal of Function Spaces
 (2018)

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## Abstract

A scalar distribution functionσ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.


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## Body

## 1. Introduction

Recall that according to the Plancherel theorem (see, e.g., [1, Ch.3.37]) the Fourier transform(1)φ^s=∫Reitsφtdtof a function φ∈L2(R) satisfies the Parseval identity(2)12π∫Rφ^s2ds=∫Rφt2dtand the inverse Fourier transform is(3)φt=12π∫Re-itsφ^sds.As is known a number of classical problems (moment problems, eigenfunctions expansion of differential operators, and their various modifications) are reduced to searching for the distribution functionsσ(·) (“spectral functions”) such that the Stieltjes integral I(λ)=∫Rgλ(s)dσ(s) with a given family of functions gλ(s) satisfies certain preassigned conditions. The problem of such a kind for the Fourier transform (1) can be formulated as follows. Assume that I=(a,b),-∞≤a<b≤∞, is an interval in R and let L2(R;I) be the set of all functions φ∈L2(R) with support in I. It is necessary to describe all distribution functions σ(·) such that instead of (2) the following Parseval identity holds:(4)∫Rφ^s2dσs=∫Rφt2dt,φ∈L2R;I.A function σ(·) satisfying (4) will be called a spectral function for the Fourier transform (1) (with respect to I); the set of all such spectral functions will be denoted by SF(I). If σ(·)∈SF(I), then for each φ∈L2(R;I) the inverse Fourier transform is(5)φt=χIt∫Re-itsφ^sdσs,where χI is the indicator of I. Clearly formulas (1), (4), and (5) give a natural generalization of the classical Plancherel theorem. We have not seen such a generalization in the literature.A parametrization of the setSF(I) in the case of a bounded interval I is given by the following theorem, which directly follows from Corollary 18 of the paper.Theorem 1.
LetI=(0,b), b<∞, be a bounded interval in R. Then the equalities(6)mKλ=i2·e-iλb+Kλe-iλb-Kλ,λ∈C+(7)σKs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡mKx+iydxestablish a bijective correspondence σ(s)=σK(s) between all holomorphic functions K(λ),λ∈C+, with Kλ≤1 and all (scalar) spectral functions σ(·)∈SF((0,b)).In the caseK(λ)≡1 the function mK(λ) is meromorphic with poles on R, σK(s) is a jump function, and equalities (1) and (5) give an expansion of a function φ∈L2(R;(0,b)) into the Fourier series on (0,b). In the case K(λ)≡0 one has σK(s)=1/2πs and equality (5) turns into the classical inverse Fourier–Plancherel transform (3) of a function φ∈L2(R) with support on (0,b). Moreover, according to Theorem 1 there exist infinitely many spectral functions σ(·)∈SF((0,b)). At the same time we show that in the case I=R the set SF(R) consists of the unique spectral function σ(s)=1/2πs, for which equality (5) turns into the inverse Fourier–Plancherel transform (3) of a function φ∈L2(R). Similar situation takes place in the case I=(0,∞) (see Corollary 22).Actually we study a more general object—a matrix-valued spectral function for the vector-valued Fourier transform. Namely, assume thatI~={I1,I2,…,In} is a system of intervals Ij=(aj,bj), -∞≤aj<bj≤∞. Denote by L2(R,Cn;I~) the set of all vector-functions (8)φt=φ1t,φ2t,…,φnt∈Cn,t∈R, such that ∫Rφ(t)2dt<∞ and support of a coordinate function φj lies in Ij. For each function φ∈L2(R,Cn;I~) equality (1) defines the vector-valued Fourier transform φ^:R→Cn of φ. A matrix-valued distribution function σ:R→Cn×n will be called a spectral function for the vector-valued Fourier transform (1) (with respect to I~) if the following Parseval identity holds: (9)∫Rdσsφ^s,φ^s=∫Rφt2dt,φ∈L2R,Cn;I~. We denote the set of all such spectral functions by SFn(I~)=SFn(I1,I2,…,In). If σ(·)∈SFn(I~), then for each φ∈L2(R,Cn;I~) the inverse Fourier transform is (10)φt=χI~t∫Re-itsdσsφ^s, where χI~(t)=diag⁡χI1t,χI2t,…,χInt (see Proposition 12).In the paper we parameterize spectral functionsσ(·)∈SFn(I~) for various classes of I~. To illustrate the obtained results we specify below two theorems, which give a parametrization of the classes SF2((0,b),(-c,0)) (b,c∈(0,∞)) and SF2((0,∞),(-∞,0)). These theorems are immediate from Corollary 19 and Theorem 21 of the paper.Theorem 2.
Assume thatb,c∈(0,∞) and let (11)Q1λ=02eiλc-20,Q2λ=-2e-iλb002,λ∈C+, where C+ is the upper half-plane of the complex plane. Then the equality (12)mKλ=-i2KλQ1λ+Q2λ-1KλQ1λ-Q2λ,λ∈C+ together with formula (7) gives a bijective correspondence σ(·)=σK(·) between all holomorphic contractive matrix functions K(λ)(∈C2×2),λ∈C+, and all spectral functions σ(·)∈SF2((0,b),(-c,0)).Theorem 3.
LetSR be the set of all complex-valued functions F on R admitting the representation (13)Fx≔limy→+0Kx+iya.e.  onR with a holomorphic function K(·) defined on C+ and satisfying Kλ≤1, λ∈C+. Then the equalities(14)ΣFx=12π1Fx¯Fx1,x∈R,andσFs=∫0sΣFxdxgive a bijective correspondence σ(·)=σF(·) between all functions F∈SR and all spectral functions σ(·)∈SF2((0,∞),(-∞,0)).These theorems show that the sets of spectral functions differ significantly for systemsI~ formed by finite or infinite intervals. In particular, by Theorem 3 each spectral function σ(·)∈SF2((0,∞),(-∞,0)) is absolutely continuous with the matrix density ΣF(x) defined by the first equality in (14).In conclusion note that the results of the paper on spectral functions for the classical Fourier transform are obtained with the aid of the results from [2–4] concerning spectral functions for canonical differential systems.


## 2. Preliminaries

### 2.1. Notations and Definitions

In the followingR+≔[0,∞) and R-=(-∞,0] are semiaxis of the real axis, C+ is an upper half-plane of the complex plane, m,n¯≔{k∈N∪{0}:m≤k≤n}(m,n∈N∪{0}andm≤n), δjk is the Kronecker delta (i.e., δjk=1 if j=k and δjk=0 if j≠k), and B(H1,H2) is the set of all linear bounded operators from the Hilbert space H1 to the Hilbert space H2, B(H)=B(H,H).We denote byS(Ck,Cn) the set of all holomorphic operator-functions K:C+→B(Ck,Cn) such that K(λ)≤1,λ∈C+ (the latter means that the function K is contractive). Moreover, we put S(Cn)≔S(Cn,Cn) and S≔S(C) (that is, S is the set of all scalar contractive holomorphic functions on C+). If K∈S(Cn), then according to the Fatou’s theorem there exists the limit(15)Fx≔limy→+0⁡Kx+iya.e.  onR.We denote by SR(Cn) the set of all functions F:R→B(Cn) admitting the representation (15) with some function K∈S(Cn). Clearly, each function K∈SR(Cn) is contractive and Borel measurable.LetI=(a,b),-∞≤a<b≤∞, be an open interval in R. We denote by L2(I,Cn) the Hilbert space of Borel measurable functions f:I→Cn such that ∫If(t)2dt<∞.Recall that a nondecreasing left-continuous operator-functionσ:R→B(Cn) with σ(0)=0 is called a distribution function. For each distribution function σ there exists a Borel measure μ on R and an operator-function Ψ:R→B(Cn) such that Ψ(s)≥0 (a.e. on R) and σ(β)-σ(α)=∫[α,β)Ψ(s)dμ, -∞<α<β<∞. Moreover, a linear space L2(σ,Cn) of all Borel measurable functions g:I→Cn satisfying ∫R(dσ(s)g(s),g(s))≔∫R(Ψ(s)g(s),g(s))dμ<∞ is a Hilbert space with respect to the inner product(16)g1,g2L2σ,Cn=∫Rdσsg1s,g2s≔∫RΨsg1s,g2sdμdefined for all g1,g2∈L2(σ,Cn) (see, e.g., [5, Ch.3.15]).A distribution functionσ is called absolutely continuous if there exists a locally integrable operator-function Σ:R→B(Cn) (the density of σ) such that σ(s)=∫0sΣ(x)dx. As is known, in this case σ′(s)=Σ(s) (a.e. on R).Remark 4.
LetCn×k be the set of all n×k-matrices with complex entries. As is known each operator T∈B(Ck,Cn) can be identified with its matrix T=(Tij)∈Cn×k in the standard bases of Ck and Cn. Hence a (holomorphic) operator-function with values in B(Ck,Cn) can be identified with a (holomorphic) matrix function with values in Cn×k, which implies that the class S(Ck,Cn) is in fact the well-known Schur class Sn×k of holomorphic in C+ contractive matrix functions with values in Cn×k. Nevertheless it is more convenient to use the operator terminology in our considerations (of course, all final results of the paper can be readily translated into the matrix language).LetA be a closed densely defined symmetric operator in H with the domain dom⁡A and let Nλ(A)=ker⁡(A∗-λ), λ∈C±, be the defect subspace of A. We will need the following definitions.Definition 5 (see [6]).
A collectionΠ={H,Γ0,Γ1} consisting of a Hilbert space H and linear mappings Γj:dom⁡A∗→Hj, j∈{0,1}, is called a boundary triplet for A∗ if the mapping Γ=(Γ0,Γ1)⊤ is surjective and the following Green’s identity holds: (17)A∗f,g-f,A∗g=Γ1f,Γ0g-Γ0f,Γ1g,f,g∈dom⁡A∗.Definition 6 (see [7, 8]).
The operator-functionM:C+→B(H) defined by(18)Γ1↾NλA=MλΓ0↾NλA,λ∈C+is called the Weyl function of the boundary triplet Π={H,Γ0,Γ1} for A∗.


### 2.2. Nevanlinna Functions and Nevanlinna Pairs

As is known a holomorphic operator-functionΦ:C+→B(Cn) is called a Nevanlinna function (and is referred to the class R[Cn]) if Im⁡Φ(λ)≥0,λ∈C+. Recall also (see, e.g., [9]) that a pair C0(·),C1(·) of holomorphic operator-functions Cj(·):C+→B(Cn),j∈{0,1}, is called a Nevanlinna pair if Im⁡C1λC0∗λ≥0 and ran⁡(C0(λ),C1(λ))=Cn,λ∈C+. Two Nevanlinna pairs C0(·),C1(·) and C~0(·),C~1(·) are said to be equivalent if C~j(λ)=φ(λ)Cj(λ),λ∈C+,j∈{0,1}, with some holomorphic operator-function φ:C+→B(Cn), such that the operator φ(λ) is invertible for all λ∈C+. Clearly, the set of all Nevanlinna pairs falls into nonintersecting classes of equivalent pairs (the equivalence classes). In the following we do not distinguish equivalent Nevanlinna pairs. Moreover, we denote by τ={C0(·),C1(·)} the equivalence class generating by the Nevanlinna pair C0(·),C1(·) and by R~(Cn) the set of all such equivalence classes. Identifying of a function Φ∈R[Cn] with a pair τ={Φ(·),-I}∈R~(Cn) enables one to consider the class R[Cn] as a subclass of R~(Cn).Definition 7 (see [3]).
LetH be a subspace in Cn and let H′=Cn⊖H. A pair τ={C0(·),C1(·)}∈R~(Cn) is referred to the class R~H(Cn) if C1(λ)↾H′=2iC0(λ)↾H′,λ∈C+.Clearly,R~Cn(Cn)=R~(Cn).A parametrization of the classR~H(Cn) in terms of contractive operator-functions is given by the following proposition.Proposition 8 (see [3]).
LetH be a subspace in Cn and let H′=Cn⊖H.Then the equalities (19)C0λ=12IH′-K1λ0K2λ+iIH:H′⊕H→H′⊕HC1λ=i2IH′iK1λ0-iK2λ-IH:H′⊕H→H′⊕H,λ∈C+give a bijective correspondence between all operator-functions K(·)∈S(H,Cn) with the block representation K(λ)=(K1(λ),K2(λ))⊤:H→H⊕H′ and all pairs τ={C0(·),C1(·)}∈R~H(Cn).


### 2.3. Canonical Systems

As is known (see, e.g., [10, 11]) a canonical differential system of an order n on the semiaxis R+=[0,∞) is of the form(20)Jy′=λHty,t∈R+,λ∈C,where J∈B(Cn) is an operator satisfying J∗=J-1=-J and H(·):R+→B(Cn) is a Borel measurable operator-function integrable on each bounded interval (0,β)⊂R+ and satisfying H(t)≥0 (a.e. on R+). In the following we assume also that H(t) is invertible a.e. on I.Denote byLH2(R+,Cn) the Hilbert space of all Borel measurable functions f:R+→Cn satisfying ∫R+Htft,ftdt<∞ and by L˙H2(R+,Cn) the set of all functions f∈LH2(R+,Cn)) such that H(t)f(t)=0 a.e. on R+∖0,βf, where βf∈(0,∞) depends on f. Since H(t) is invertible a.e. on R+, system (20) gives rise to the maximal operator Tmax and the minimal operator Tmin in LH2(R+,Cn) defined as follows. Let AC(R+,Cn) be the set of all functions f:R+→Cn, which are absolutely continuous on each compact subinterval [0,β]⊂R+. Then the domain dom⁡Tmax of Tmax is dom⁡Tmax={y∈AC(R+,Cn)∩LH2(R+,Cn):H-1tJy′(t)∈LH2(R+,Cn)} and Tmaxy=H-1tJy′(t),y∈dom⁡Tmax. The minimal operator Tmin is the closure of the operator Tmin′ defined as a restriction of Tmax onto the set of all functions y∈dom⁡Tmax with compact support. It turns out that Tmin is a closed densely defined symmetric operator, Tmin∗=Tmax, and there exists the limit(21)y,z∞≔limt→+∞Jyt,zt,y,z∈dom⁡Tmax(see, e.g., [12]). Clearly, equality (21) defines a skew-Hermitian bilinear form [·,·]∞ on dom⁡Tmax.For eachλ∈C there exists a unique locally absolutely continuous operator-function Y(·,λ):R+→B(Cn) such that JYt′(t,λ)=λH(t)Y(t,λ) (a.e. on R+) and Y(0,λ)=In. The operator-function Y(·,λ) is called the matrizant of system (20).In the following we denote byN± the deficiency indices of the system (20), i.e., deficiency indices of Tmin. Clearly, N±=dim⁡Nλ,λ∈C±, where Nλ⊂dom⁡Tmax is the linear space of all vector solutions of (20) belonging to LH2(R+,Cn). Hence N±≤n. Moreover, if N+=n, then Y(·,λ)x∈dom⁡Tmax(x∈Cn,λ∈C+) and hence for any λ∈C+ the equality(22)Yλxt=Yt,λx,x∈Cndefines a linear operator Y(λ):Cn→dom⁡Tmax.With each functionf∈L˙H2(R+,Cn) one associates the function f^:R→Cn (the generalized Fourier transform of f) given by(23)f^s=∫R+Y∗t,sHtftdt,s∈R.Definition 9.
A distribution functionσ^(·):R→B(Cn) is called a spectral function of system (20) if the following Parseval identity holds:(24)∫Rdσ^sf^s,f^s=∫R+Htft,ftdt,f∈L˙H2R+,Cn.Below within this subsection we assume thatn=2q+ν (hence Cn=Cq⊕Cν⊕Cq) and the operator J in (20) is(25)J=00-Iq0iIν0Iq00:Cq⊕Cν⊕Cq→Cq⊕Cν⊕Cq.Assume that system (20) has the maximal deficiency index N+=n and let m-=n-N-. Then m-≤q, and by [3, Lemma 4.1] there exists an operator(26)Γ∞=Γ0∞,Γ^∞,Γ1∞⊤:dom⁡Tmax→Cq⊕Cν⊕Cqsuch that(27)ran⁡Γ∞=x⊕x^⊕x′∈Cq⊕Cν⊕Cq:Px′=-i2Px(28)y,z∞=Γ0∞y,Γ1∞z-Γ1∞y,Γ0∞z+iΓ^∞y,Γ^∞z,y,z∈dom⁡Tmax(here P is the orthoprojector in Cq onto Cm-).A description of all spectral functions of system (20) is given by the following theorem, which directly follows from Theorem 4.17 and formulas (4.68)–(4.71) in [3].Theorem 10.
Let system (20), (25) satisfies N+=n, let m-=n-N-, let H=Cq-m-⊕Cν⊕Cq(⊂Cn), let Γ∞ be the operator (26) satisfying (27), (28), and let Y(λ) be the operator (22). Moreover, let B:C+→B(Cn) be an operator-function (the monodromy matrix) defined by B(λ)=Γ∞Y(λ),λ∈C+, let(29)Bλ=B11λB12λB13λB21λB22λB23λB31λB32λB33λ:Cq⊕Cν⊕Cq→Cq⊕Cν⊕Cqbe the block representation ofB(λ), and let(30)w1λ=B11λB12λB13λ-iB21λ-iB22λ-Iν-iB23λ00-Iq(31)w2λ=12B13λi2B12λ-12B11λ-i2B23λ12B22λ+Iνi2B21λ12Iq00(32)w3λ=B31λB32λB33λ-12B21λ-12B22λ+Iν-12B23-Iq00(33)w4λ=12B33λi2B32λ-12B31λ-14B23λ-i4B22λ-Iν14B21λ00-12Iq.Then the equalities(34)m^τλ=C0λw1λ+C1λw3λ-1C0λw2λ+C1λw4λ,λ∈C+(35)σ^τs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡m^τx+iydxestablish a bijective correspondence σ^(s)=σ^τ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ^(·) of the system.


## 2.1. Notations and Definitions

In the followingR+≔[0,∞) and R-=(-∞,0] are semiaxis of the real axis, C+ is an upper half-plane of the complex plane, m,n¯≔{k∈N∪{0}:m≤k≤n}(m,n∈N∪{0}andm≤n), δjk is the Kronecker delta (i.e., δjk=1 if j=k and δjk=0 if j≠k), and B(H1,H2) is the set of all linear bounded operators from the Hilbert space H1 to the Hilbert space H2, B(H)=B(H,H).We denote byS(Ck,Cn) the set of all holomorphic operator-functions K:C+→B(Ck,Cn) such that K(λ)≤1,λ∈C+ (the latter means that the function K is contractive). Moreover, we put S(Cn)≔S(Cn,Cn) and S≔S(C) (that is, S is the set of all scalar contractive holomorphic functions on C+). If K∈S(Cn), then according to the Fatou’s theorem there exists the limit(15)Fx≔limy→+0⁡Kx+iya.e.  onR.We denote by SR(Cn) the set of all functions F:R→B(Cn) admitting the representation (15) with some function K∈S(Cn). Clearly, each function K∈SR(Cn) is contractive and Borel measurable.LetI=(a,b),-∞≤a<b≤∞, be an open interval in R. We denote by L2(I,Cn) the Hilbert space of Borel measurable functions f:I→Cn such that ∫If(t)2dt<∞.Recall that a nondecreasing left-continuous operator-functionσ:R→B(Cn) with σ(0)=0 is called a distribution function. For each distribution function σ there exists a Borel measure μ on R and an operator-function Ψ:R→B(Cn) such that Ψ(s)≥0 (a.e. on R) and σ(β)-σ(α)=∫[α,β)Ψ(s)dμ, -∞<α<β<∞. Moreover, a linear space L2(σ,Cn) of all Borel measurable functions g:I→Cn satisfying ∫R(dσ(s)g(s),g(s))≔∫R(Ψ(s)g(s),g(s))dμ<∞ is a Hilbert space with respect to the inner product(16)g1,g2L2σ,Cn=∫Rdσsg1s,g2s≔∫RΨsg1s,g2sdμdefined for all g1,g2∈L2(σ,Cn) (see, e.g., [5, Ch.3.15]).A distribution functionσ is called absolutely continuous if there exists a locally integrable operator-function Σ:R→B(Cn) (the density of σ) such that σ(s)=∫0sΣ(x)dx. As is known, in this case σ′(s)=Σ(s) (a.e. on R).Remark 4.
LetCn×k be the set of all n×k-matrices with complex entries. As is known each operator T∈B(Ck,Cn) can be identified with its matrix T=(Tij)∈Cn×k in the standard bases of Ck and Cn. Hence a (holomorphic) operator-function with values in B(Ck,Cn) can be identified with a (holomorphic) matrix function with values in Cn×k, which implies that the class S(Ck,Cn) is in fact the well-known Schur class Sn×k of holomorphic in C+ contractive matrix functions with values in Cn×k. Nevertheless it is more convenient to use the operator terminology in our considerations (of course, all final results of the paper can be readily translated into the matrix language).LetA be a closed densely defined symmetric operator in H with the domain dom⁡A and let Nλ(A)=ker⁡(A∗-λ), λ∈C±, be the defect subspace of A. We will need the following definitions.Definition 5 (see [6]).
A collectionΠ={H,Γ0,Γ1} consisting of a Hilbert space H and linear mappings Γj:dom⁡A∗→Hj, j∈{0,1}, is called a boundary triplet for A∗ if the mapping Γ=(Γ0,Γ1)⊤ is surjective and the following Green’s identity holds: (17)A∗f,g-f,A∗g=Γ1f,Γ0g-Γ0f,Γ1g,f,g∈dom⁡A∗.Definition 6 (see [7, 8]).
The operator-functionM:C+→B(H) defined by(18)Γ1↾NλA=MλΓ0↾NλA,λ∈C+is called the Weyl function of the boundary triplet Π={H,Γ0,Γ1} for A∗.


## 2.2. Nevanlinna Functions and Nevanlinna Pairs

As is known a holomorphic operator-functionΦ:C+→B(Cn) is called a Nevanlinna function (and is referred to the class R[Cn]) if Im⁡Φ(λ)≥0,λ∈C+. Recall also (see, e.g., [9]) that a pair C0(·),C1(·) of holomorphic operator-functions Cj(·):C+→B(Cn),j∈{0,1}, is called a Nevanlinna pair if Im⁡C1λC0∗λ≥0 and ran⁡(C0(λ),C1(λ))=Cn,λ∈C+. Two Nevanlinna pairs C0(·),C1(·) and C~0(·),C~1(·) are said to be equivalent if C~j(λ)=φ(λ)Cj(λ),λ∈C+,j∈{0,1}, with some holomorphic operator-function φ:C+→B(Cn), such that the operator φ(λ) is invertible for all λ∈C+. Clearly, the set of all Nevanlinna pairs falls into nonintersecting classes of equivalent pairs (the equivalence classes). In the following we do not distinguish equivalent Nevanlinna pairs. Moreover, we denote by τ={C0(·),C1(·)} the equivalence class generating by the Nevanlinna pair C0(·),C1(·) and by R~(Cn) the set of all such equivalence classes. Identifying of a function Φ∈R[Cn] with a pair τ={Φ(·),-I}∈R~(Cn) enables one to consider the class R[Cn] as a subclass of R~(Cn).Definition 7 (see [3]).
LetH be a subspace in Cn and let H′=Cn⊖H. A pair τ={C0(·),C1(·)}∈R~(Cn) is referred to the class R~H(Cn) if C1(λ)↾H′=2iC0(λ)↾H′,λ∈C+.Clearly,R~Cn(Cn)=R~(Cn).A parametrization of the classR~H(Cn) in terms of contractive operator-functions is given by the following proposition.Proposition 8 (see [3]).
LetH be a subspace in Cn and let H′=Cn⊖H.Then the equalities (19)C0λ=12IH′-K1λ0K2λ+iIH:H′⊕H→H′⊕HC1λ=i2IH′iK1λ0-iK2λ-IH:H′⊕H→H′⊕H,λ∈C+give a bijective correspondence between all operator-functions K(·)∈S(H,Cn) with the block representation K(λ)=(K1(λ),K2(λ))⊤:H→H⊕H′ and all pairs τ={C0(·),C1(·)}∈R~H(Cn).


## 2.3. Canonical Systems

As is known (see, e.g., [10, 11]) a canonical differential system of an order n on the semiaxis R+=[0,∞) is of the form(20)Jy′=λHty,t∈R+,λ∈C,where J∈B(Cn) is an operator satisfying J∗=J-1=-J and H(·):R+→B(Cn) is a Borel measurable operator-function integrable on each bounded interval (0,β)⊂R+ and satisfying H(t)≥0 (a.e. on R+). In the following we assume also that H(t) is invertible a.e. on I.Denote byLH2(R+,Cn) the Hilbert space of all Borel measurable functions f:R+→Cn satisfying ∫R+Htft,ftdt<∞ and by L˙H2(R+,Cn) the set of all functions f∈LH2(R+,Cn)) such that H(t)f(t)=0 a.e. on R+∖0,βf, where βf∈(0,∞) depends on f. Since H(t) is invertible a.e. on R+, system (20) gives rise to the maximal operator Tmax and the minimal operator Tmin in LH2(R+,Cn) defined as follows. Let AC(R+,Cn) be the set of all functions f:R+→Cn, which are absolutely continuous on each compact subinterval [0,β]⊂R+. Then the domain dom⁡Tmax of Tmax is dom⁡Tmax={y∈AC(R+,Cn)∩LH2(R+,Cn):H-1tJy′(t)∈LH2(R+,Cn)} and Tmaxy=H-1tJy′(t),y∈dom⁡Tmax. The minimal operator Tmin is the closure of the operator Tmin′ defined as a restriction of Tmax onto the set of all functions y∈dom⁡Tmax with compact support. It turns out that Tmin is a closed densely defined symmetric operator, Tmin∗=Tmax, and there exists the limit(21)y,z∞≔limt→+∞Jyt,zt,y,z∈dom⁡Tmax(see, e.g., [12]). Clearly, equality (21) defines a skew-Hermitian bilinear form [·,·]∞ on dom⁡Tmax.For eachλ∈C there exists a unique locally absolutely continuous operator-function Y(·,λ):R+→B(Cn) such that JYt′(t,λ)=λH(t)Y(t,λ) (a.e. on R+) and Y(0,λ)=In. The operator-function Y(·,λ) is called the matrizant of system (20).In the following we denote byN± the deficiency indices of the system (20), i.e., deficiency indices of Tmin. Clearly, N±=dim⁡Nλ,λ∈C±, where Nλ⊂dom⁡Tmax is the linear space of all vector solutions of (20) belonging to LH2(R+,Cn). Hence N±≤n. Moreover, if N+=n, then Y(·,λ)x∈dom⁡Tmax(x∈Cn,λ∈C+) and hence for any λ∈C+ the equality(22)Yλxt=Yt,λx,x∈Cndefines a linear operator Y(λ):Cn→dom⁡Tmax.With each functionf∈L˙H2(R+,Cn) one associates the function f^:R→Cn (the generalized Fourier transform of f) given by(23)f^s=∫R+Y∗t,sHtftdt,s∈R.Definition 9.
A distribution functionσ^(·):R→B(Cn) is called a spectral function of system (20) if the following Parseval identity holds:(24)∫Rdσ^sf^s,f^s=∫R+Htft,ftdt,f∈L˙H2R+,Cn.Below within this subsection we assume thatn=2q+ν (hence Cn=Cq⊕Cν⊕Cq) and the operator J in (20) is(25)J=00-Iq0iIν0Iq00:Cq⊕Cν⊕Cq→Cq⊕Cν⊕Cq.Assume that system (20) has the maximal deficiency index N+=n and let m-=n-N-. Then m-≤q, and by [3, Lemma 4.1] there exists an operator(26)Γ∞=Γ0∞,Γ^∞,Γ1∞⊤:dom⁡Tmax→Cq⊕Cν⊕Cqsuch that(27)ran⁡Γ∞=x⊕x^⊕x′∈Cq⊕Cν⊕Cq:Px′=-i2Px(28)y,z∞=Γ0∞y,Γ1∞z-Γ1∞y,Γ0∞z+iΓ^∞y,Γ^∞z,y,z∈dom⁡Tmax(here P is the orthoprojector in Cq onto Cm-).A description of all spectral functions of system (20) is given by the following theorem, which directly follows from Theorem 4.17 and formulas (4.68)–(4.71) in [3].Theorem 10.
Let system (20), (25) satisfies N+=n, let m-=n-N-, let H=Cq-m-⊕Cν⊕Cq(⊂Cn), let Γ∞ be the operator (26) satisfying (27), (28), and let Y(λ) be the operator (22). Moreover, let B:C+→B(Cn) be an operator-function (the monodromy matrix) defined by B(λ)=Γ∞Y(λ),λ∈C+, let(29)Bλ=B11λB12λB13λB21λB22λB23λB31λB32λB33λ:Cq⊕Cν⊕Cq→Cq⊕Cν⊕Cqbe the block representation ofB(λ), and let(30)w1λ=B11λB12λB13λ-iB21λ-iB22λ-Iν-iB23λ00-Iq(31)w2λ=12B13λi2B12λ-12B11λ-i2B23λ12B22λ+Iνi2B21λ12Iq00(32)w3λ=B31λB32λB33λ-12B21λ-12B22λ+Iν-12B23-Iq00(33)w4λ=12B33λi2B32λ-12B31λ-14B23λ-i4B22λ-Iν14B21λ00-12Iq.Then the equalities(34)m^τλ=C0λw1λ+C1λw3λ-1C0λw2λ+C1λw4λ,λ∈C+(35)σ^τs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡m^τx+iydxestablish a bijective correspondence σ^(s)=σ^τ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ^(·) of the system.


## 3. Spectral Functions for the Fourier Transform

### 3.1. Definitions and Auxiliary Results

Let(36)I~=I1,I2,…,Inbe a system of intervals Ij=(aj,bj), -∞≤aj<bj≤∞, j∈1,n¯. Denote by L2(R,Cn;I~) the set of all functions φ(·)={φj(·)}j=1n∈L2(R,Cn) with φj(t)=0 a.e. on R∖Ij and by L˙2(R,Cn;I~) the set of all functions φ(·)={φj(·)}j=1n∈L2(R,Cn;I~) with the following property: if Ij∈I~ is an infinite interval, then there exists a finite interval (αj,βj)⊂Ij (depending on φ) such that φj(t)=0 a.e. on Ij∖(αj,βj). Since L2(R,Cn;I~) is unitarily equivalent to the Hilbert space L2(I1)⊕L2(I2)⊕⋯⊕L2(In), it follows that L2(R,Cn;I~) is a closed subspace in L2(R,Cn). Moreover, L˙2(R,Cn;I~) is a linear manifold dense in L2(R,Cn;I~).With each functionφ∈L˙2(R,Cn;I~) one associates the function φ^:R→Cn (the classical vector-valued Fourier transform of φ) given by(37)φ^s=∫Reitsφtdt,s∈R.Assume that σ(·):R→B(Cn) is an operator-valued distribution function such that for each φ∈L˙2(R,Cn;I~) the following Parseval identity holds:(38)φ^L2σ;Cn≔∫Rdσsφ^s,φ^s=∫Rφt2dt≕φL2R,Cn.Then for each φ∈L2(R,Cn;I~) the integral in (37) converges in the norm of L2(σ,Cn) and the equality Vσφ=φ^,φ∈L2(R,Cn;I~), defines an isometry Vσ from L2(R,Cn;I~) to L2(σ;Cn).Definition 11.
AB(Cn)-valued distribution function σ(·) such that (38) holds will be called a spectral function for the classical Fourier transform (37) (with respect to the system of intervals I~).In what follows the set of spectral functions in the sense of Definition11 will be denoted by SFn(I~)=SFn(I1,I2,…,In). Moreover, in the case when n=1 and the system I~ consists of a unique interval I=(a,b), we let SF(a,b)≔SF1(I~).Proposition 12.
LetI~ be system of intervals (36), let χIj(·) be the indicator of an interval Ij, let χI~(t)=diag⁡(χI1(t),χI2(t),…,χIn(t)), and let σ(·)∈SFn(I~). Then the inverse to (37) transform is(39)φt=∫Re-itsχI~tdσsφ^s,where the integral converges in the norm of L2(R,Cn). If in addition σ(·) is absolutely continuous and Σ(s)=σ′(s) is the density of σ(·), then Parseval identity (38) and the inverse transform (39) can be written as(40)∫RΣsφ^s,φ^sds=∫Rφt2dt,φ∈L˙2R,Cn;I~(41)φt=∫Re-itsχI~tΣsφ^sds.Proof.
SinceVσ is an isometry, to prove (39) it is sufficient to show that(42)Vσ∗gt=∫Re-itsχI~tdσsgs,g∈L˙2σ,Cn,where L˙2(σ,Cn) is the set of all functions g∈L2(σ,Cn) with compact support.
Letg∈L˙2(σ,Cn), let fg(t)={fg,j(t)}j=1n be given by (43)fgt=∫Re-itsχI~tdσsgs≔χI~t∫Re-itsΨsgsdμ (for Ψ and μ see assertions before (16)), and let f∗(t)={fj∗(t)}j=1n≔(Vσ∗g)(t)(∈L2(R,Cn;I~)). Let us show that for each finite interval Ij′⊂Ij(44)∫RχIj′tfg,jt¯dt=∫RχIj′tfj∗t¯dt,j∈1,n¯.Indeed, let αj(t)={δkjχIj′(t)}k=1n∈L2(R,Cn;I~). Then by using the Fubini theorem and (16) one gets (45)∫RχIj′tfg,jt¯dt=∫Rαjt,fgtdt=∫Rαjt,∫Re-itsΨsgsdμdt=∫R∫Rαjt,e-itsΨsgsdμdt=∫R∫Rαjt,e-itsΨsgsdtdμ=∫RΨs∫Reitsαjtdt,gsdμ=Vσαj,gL2σ;Cn=αj,Vσ∗gL2R,Cn;I~=∫Rαjt,f∗tdt=∫RχIj′tfj∗t¯dt,which proves (44). Since fg,j(t)=fj∗(t)=0 (a.e. on R∖Ij), it follows from (44) that fg(t)=f∗(t) (a.e. on R) and hence (42) holds.
The last statement of the proposition is obvious.Remark 13.
Ifn=1 and system I~ consists of a unique interval I=(a,b), then equality (37) defines the classical C-valued Fourier transform φ^ of a scalar function φ∈L˙2(R) with compact support belonging to (a,b) (we denote the set of such functions by L˙2(R;(a,b))). A spectral function σ(·)∈SF(a,b) for this transform is a scalar distribution function such that the Parseval identity (46)∫Rφ^s2dσs=∫Rφt2dt,φ∈L˙2R;a,bholds; moreover, the inverse Fourier transform is (47)φt=∫Re-itsχItφ^sdσs.Assume thatr±,m±∈N∪{0}, p≔r++m+,q≔r-+m-, n≔p+q, and I~ is a system of intervals given by(48)I~=0,b1,…,0,br+,R+,…,R+︸m+times︷p,-c1,0,…,-cr-,0,R-,…,R-︸m-times︷q(here bj,cj∈(0,∞)). With the system I~ one associates canonical differential system(49)J~y′=λH~ty,t∈R+,where(50)J~=idiag⁡Ip,-Iq,H~t=diag⁡h1+t,…,hp+t,h1-t,…,hq-t.In (50) hj+,j∈1,p¯, and hk-,k∈1,q¯, are continuous functions on R+ such that hj+(t)>0, hk-(t)>0, t∈R+, and(51)∫R+hj+tdt=bj,j∈1,r+¯;∫R+hj+tdt=+∞,j∈r++1,p¯(52)∫R+hk-tdt=ck,k∈1,r-¯;∫R+hk-tdt=+∞,k∈r-+1,q¯.One can easily verify that the matrizant Y~(·,λ) of system (49) is(53)Y~t,λ=diag⁡e-iλΦ1+t,…,e-iλΦp+t,eiλΦ1-t,…,eiλΦq-t,where(54)Φj+t=∫0,thj+udu,j∈1,p¯;Φk-t=∫0,thk-udu,k∈1,q¯.Assume that εj={δkj}k=1n∈Cn, j∈1,n¯, is the standard basis in Cn and let(55)y~j+t,λ=e-iλΦj+tεj,j∈1,p¯;y~k-t,λ=eiλΦk-tεp+k,k∈1,q¯.Since by (53) (y~j+(t,λ))⊤ and (y~k-(t,λ))⊤ are columns of the matrizant Y~(t,λ), the system {y~j+(·,λ)}j=1p∪{y~k-(·,λ)}k=1q is a basis in the space of all solutions of system (49). Moreover, the immediate checking shows that(56)y~j+·,λ∈LH~2R+,Cn,j∈1,r+¯;y~j+·,λ∉LH~2R+,Cn,j∈r++1,p¯(57)y~k-·,λ∈LH~2R+,Cn,k∈1,q¯for λ∈C+ and(58)y~j+·,λ∈LH~2R+,Cn,j∈1,p¯;y~k-·,λ∈LH~2R+,Cn,k∈1,r-¯(59)y~k-·,λ∉LH~2R+,Cn,k∈r-+1,q¯for λ∈C-. Therefore deficiency indices N~± of system (49) are(60)N~+=q+r+,N~-=p+r-.Now assume that I~ is system of intervals (48) with p≥q and let ν≔p-q, so that (61)p=q+ν=r-+m-+ν,n=p+q=2q+ν. Moreover, let X and X∗ be mutually adjoint unitary operators in Cn defined by(62)X=12Iq0iIq02Iν0Iq0-iIq,X∗=12Iq0Iq02Iν0-iIq0iIq.Together with canonical system (49), (50) we associate with system of intervals I~ canonical system (20), where J is of the form (25) and(63)Ht=X∗H~tX,t∈R+.Clearly, J~ admits the representation J~=diag⁡iIq,iIν,-iIq and the immediate checking shows that(64)J=X∗J~X,J~=XJX∗.In the following we denote by Tmax and T~max the maximal operators induced by systems (20) and (49), respectively. It follows from (63) and (64) that the equality(65)Uf~t=X∗f~t,f~∈LH~2R+,Cndefines a unitary operator U from LH~2(R+,Cn) onto LH2(R+,Cn) such that Udom⁡T~max=dom⁡Tmax and(66)JUy~t,Uz~t=J~y~t,z~t,Uy~,Uz~∞=y~,z~∞,y~,z~∈dom⁡T~max.Moreover, the matrizant Y(·,λ) of system (20) is(67)Yt,λ=X∗Y~t,λX,where Y~(t,λ) is the matrizant (53) of system (49). Observe also that systems (20) and (49) have the same deficiency indices and by (60) deficiency indices N± of the system (20) are(68)N+=q+r+,N-=p+r-.Proposition 14.
LetI~ be system of intervals (48). Then the set SFn(I~) coincides with the set of spectral functions of system (49), (50). If in addition p≥q, then the equality(69)σs=Xσ^sX∗gives a bijective correspondence between all spectral functions σ^(·) of system (20), (63) and all spectral functions σ(·)∈SFn(I~).Proof.
LetIj+=[0,bj), j∈1,r+¯, and Ij+=R+,j∈r++1,p¯. Moreover, let Ik-=-ck,0, k∈1,r-¯, and Ik-=R-, k∈r-+1,q¯. Clearly, Φj+(·) (Φk-(·)) is a strictly increasing absolutely continuous function mapping bijectively R+ onto Ij+ (resp. -Ik-={t∈R:-t∈Ik-}). With each vector-function(70)f·=f1+·,…,fp+·,f1-·,…,fq-·:R+→Cnwe associate the vector-function(71)φf·=φ1+·,…,φp+·,φ1-·,…,φq-·:R→Cngiven by (72)φj+u=fj+Φj+-1u,u∈Ij+0,u∉Ij+,φk-u=fk-Φk--1-u,u∈Ik-0,u∉Ik- (here j∈1,p¯ and k∈1,q¯). Since Φj+′(t)=hj+(t) and Φk-′(t)=hk-(t), it follows that (73)∫R+hj+tfj+t2dt=∫Ij+fj+Φj+-1u2du=∫Rφj+u2du,j∈1,p¯∫R+hk-tfk-t2dt=∫Ik-fk-Φk--1-u2du=∫Rφk-u2du,k∈1,q¯ and hence (74)∫R+H~tft,ftdt=Σj=1p∫R+hj+tfj+t2dt+Σk=1q∫R+hk-tfk-t2dt=Σj=1p∫Rφj+u2du+Σk=1q∫Rφk-u2du=∫Rφfu2du.Thus, the equality φ(t)=φf(t) gives a bijective correspondence between all functions f∈L˙H~2(R+,Cn) and all functions φ∈L˙2(R,Cn;I~). Moreover, fLH~2(R+,Cn)=φfL2(R,Cn;I~).
Next assume thatf∈L˙H~2(R+,Cn) and φf∈L˙2(R,Cn;I~) are vector-functions (70) and (71) and let (75)f^=f^1+,…,f^p+,f^1-,…,f^q-,φ^f=φ^1+,…,φ^p+,φ^1-,…,φ^q- be the corresponding Fourier transforms (23) and (37) (in (23) H(t) and Y(t,λ) should be replaced with H~(t) and Y~(t,λ), respectively). Then by (53) and (50) (76)f^j+s=∫R+eisΦj+thj+tfj+tdt=∫Ij+eisufj+Φj+-1udu=∫Reisuφj+udu=φ^j+sf^k-s=∫R+e-isΦk-thk-tfk-tdt=∫Ik-eisufk-Φk--1-udu=∫Reisuφk-udu=φ^k-sand, consequently, f^(s)=φ^f(s). This yields the first statement of the proposition.
Ifp≥q, then by (63) and (67) the equality (69) gives a bijective correspondence between all spectral functions σ^(·) of system (20), (63) and all spectral functions σ(·) of system (49), (50). This and the first statement of the proposition yield the second statement.Below unless otherwise stated we assume thatI~ is system of intervals (48) with m+=0 and p≥q. This means that(77)I~=0,b1,…,0,bp,-c1,0,…,-cr-,0,R-,…,R-︸m-times︷q,and according to (68) deficiency indices N± of the respective system (20) are(78)N+=q+p=n,N-=p+r-.Lemma 15.
Letεj={δkj}k=1n,j∈1,n¯, be the standard basis in Cn, let εj′=X∗εj and let(79)ujt=eΦj+t-bjεj′,j∈1,p¯;vkt=eΦk-t-ckεp+k′k∈1,r-¯,t∈R+,where Φj+(t) and Φk-(t) are given by (54). Then uj,vk∈dom⁡Tmax and the equalities(80)G0y=y,uj∞j=1p∈Cp,G1y=y,vk∞k=1r-∈Cr-,y∈dom⁡Tmaxdefine linear mappings G0:dom⁡Tmax→Cp and G1:dom⁡Tmax→Cr- such that the mapping G≔(G0,G1)⊤:dom⁡Tmax→Cp⊕Cr- is surjective and the identity(81)y,z∞=iG0y,G0zCp-iG1y,G1zCr-,y,z∈dom⁡Tmaxis valid. Moreover, if(82)G0=G01,G02,G03⊤:dom⁡Tmax→Cr-⊕Cm-⊕Cν︸Cpis the block representation of G0, then the equalities(83)Γ0∞=12G01+G1,G02⊤:dom⁡Tmax→Cr-⊕Cm-︸Cq(84)Γ^∞=G03:dom⁡Tmax→Cν(85)Γ1∞-i2G01-G1,-i2G02⊤:dom⁡Tmax→Cr-⊕Cm-︸Cqtogether with (26) define the operator Γ∞ such that (27) and (28) hold.Proof.
Let(86)u~jt=eΦj+t-bjεj,j∈1,p¯;v~kt=eΦk-t-ckεp+k,k∈1,r-¯.Then by (55) u~j(t)=e-bjy~j+(t,i), v~k(t)=e-cky~k-(t,-i), and in view of (56) and (58) u~j,v~k∈dom⁡T~max. Since uj(t)=X∗u~j(t) and vk(t)=X∗v~k(t), it follows that(87)uj=Uu~j,j∈1,p¯;vk=Uv~k,k∈1,r-¯,where U is the unitary operator (65). Hence uj,vk∈dom⁡Tmax.
Clearly,(88)J~εj,εk=iδjk,j,k∈1,p¯0,j∈1,p¯,k∈p+1,n,¯-iδjk,j,k∈p+1,n¯.This and (66), (87) yield (89)Jujt,ult=J~u~jt,u~lt=ieΦj+t-bjeΦl+t-blδjl,j,l∈1,p¯Jujt,vkt=J~u~jt,v~kt=0,j∈1,p¯,k∈1,r-¯Jvkt,vmt=J~v~kt,v~mt=-ieΦk-t-ckeΦm-t-cmδkm,k,m∈1,r-¯. Since by (51) and (52) Φj+(∞)=bj, j∈1,p¯, and Φk-(∞)=ck, k∈1,r-¯, it follows that(90)uj,ul∞=iδjl,j,l∈1,p¯;uj,vk∞=0,j∈1,p¯,k∈1,r-¯(91)vk,vm∞=-iδkm,k,m∈1,r-¯.Next, denote by D0 the kernel of the form [·,·]∞. According to [13, Proposition 5.5] the indices of inertia ν∞± of the form [·,·]∞ are ν∞+=N+-q,ν∞-=N--p and (78) yields ν∞+=p,ν∞-=r-. Therefore by (90) and (91) system {uj}j=1p∪{vk}k=1r- forms a basis in dom⁡Tmax/D0 and hence each y∈dom⁡Tmax admits a unique representation (92)y=y0+Σj=1pαjyuj+Σk=1r-βkyvk, where y0∈D0 and αjy,βky∈C. Let G0:dom⁡Tmax→Cp and G1:dom⁡Tmax→Cr- be the operators given by(93)G0y=iαjyj=1p∈Cp,G1y=-iβkyk=1r-∈Cr-.Then by (90) and (91) for any y,z∈dom⁡Tmax one has (94)y,z∞=iΣj=1pαjyαjz¯-iΣk=1r-βkyβkz¯=iG0y,G0zCp-iG1y,G1zCr-.Moreover, by (93) the operator G=(G0,G1)⊤ is surjective. Note also that by (90) and (91) [y,uj]∞=iαjy, [y,vk]∞=-iβky, and hence (93) admits the representation (80). This proves the first statement of the lemma.
Now assume thatΓ0∞,Γ^∞ and Γ1∞ are the operators (83)–(85), Γ∞ is given by (26), and P is the orthoprojector in Cq onto Cm-. Then (95)PΓ1∞=-i2G02=-i2PΓ0∞, which in view of surjectivity of G yields (27). Finally, for any y,z∈dom⁡Tmax one has (96)Γ0∞y,Γ1∞z-Γ1∞y,Γ0∞z+iΓ^∞y,Γ^∞z=i2G01y+G1y,G01z-G1z+i2G02y,G02z+i2G01y-G1y,G01z+G1z+i2G02y,G02z+iG03y,G03z=iG01y,G01z-iG1y,G1z+iG02y,G02z+iG03y,G03z=iG0y,G0z-iG1y,G1z=y,z∞.This proves (28).


### 3.2. Parametrization of Spectral Functions

A parametrization of all spectral functionsσ(·)∈SFn(I~) is given by the following theorem.Theorem 16.
Assume thatp,r-,m-∈N∪{0}, q≔r-+m-≤p, ν≔p-q, n≔p+q, and I~ is a system of intervals (77). Let H≔Cr-⊕Cν⊕Cq, let ψj:C+→B(Cq),j∈1,3¯ and ψ4:C+→B(Cν) be operator (matrix) functions given by(97)ψ1λ=diag⁡ie-iλb1,…,ie-iλbr-,i2e-iλbr-+1,…,i2e-iλbq(98)ψ2λ=diag⁡-ieiλc1,…,-ieiλcr-,0,…,0︸m-times(99)ψ3λ=diag⁡ie-iλb1,…,ie-iλbr-,i2e-iλbr-+1,…,i2e-iλbq(100)ψ4λ=diag⁡ie-iλbq+1,…,ie-iλbp,and let(101)ω1λ=12ψ1λ0ψ2λ0-i2ψ4λ-Iν0iIq0-iIq∈BCq⊕Cν⊕Cq︸Cn(102)ω2λ=122iψ1λ0-iψ2λ02ψ4λ+Iν0Iq0Iq∈BCq⊕Cν⊕Cq︸Cn(103)ω3λ=12-iψ3λ0iψ2λ0-12ψ4λ+Iν0-Iq0-Iq∈BCq⊕Cν⊕Cq︸Cn(104)ω4λ=122ψ3λ0ψ2λ0-i2ψ4λ-Iν0iIq0-iIq∈BCq⊕Cν⊕Cq︸Cn(in formulas (97)–(104) λ∈C+). Then the equalities(105)mτλ=C0λω1λ+C1λω3λ-1C0λω2λ+C1λω4λ,λ∈C+(106)στs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡mτx+iydxestablish a bijective correspondence σ(s)=στ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ(·)∈SFn(I~).Proof.
First by using Theorem10 we parameterize spectral functions of system (20) with J of the form (25) and H(t) defined by (63) and (50).
LetΓ∞ be the operator defined by (83)–(85) and (26) and let Y(·,λ) be the matrizant of system (20). Since N+=n (see (78)) and by Lemma 15Γ∞ satisfies (27) and (28), the equality B(λ)=Γ∞Y(λ), λ∈C+, defines the monodromy matrix B(λ). To calculate B(λ) we first calculate the operators D0(λ)=G0Y(λ)∈B(Cn,Cp) and D1(λ)=G1Y(λ)∈B(Cn,Cr-),λ∈C+ (here G0 and G1 are operators (80)). Let u~j,v~k∈dom⁡T~max be given by (86) and let G~0:dom⁡T~max→Cp and G~1:dom⁡T~max→Cr- be operators defined by (107)G~0y~=y~,u~j∞j=1p∈Cp,G~1y=y~,v~k∞k=1r-∈Cr-,y~∈dom⁡T~max. Moreover, let Y~(·,λ) be the matrizant (53) of system (49), (50), let Y~(λ) be the respective operator (22), and let D~0(λ)=G~0Y~(λ)∈B(Cn,Cp), D~1(λ)=G~1Y~(λ)∈B(Cn,Cr-),λ∈C+. Since (y~j+)⊤ and (y~k-)⊤ (see (55)) are columns of the matrizant (53), it follows that (108)D~mλ=G~my~1+⊤,…,G~my~p+⊤,G~my~1-⊤,…,G~my~q-⊤,m∈0,1. Moreover, by (88) (109)y~j+,u~l∞=limt→+∞⁡J~y~j+t,λ,u~lt=limt→+∞⁡ie-iλΦj+teΦl+t-blδjl=ie-iλbjδjl,j,l∈1,p¯y~j+,v~k∞=limt→+∞⁡J~y~j+t,λ,v~kt=0,j∈1,p¯,k∈1,r-¯y~k-,u~j∞=limt→+∞⁡J~y~k-t,λ,u~jt=0,k∈1,q¯,j∈1,p¯y~k-,v~m∞=limt→+∞⁡J~y~k-t,λ,v~mt=limt→+∞⁡-ieiλΦk-teΦm-t-cmδkm=-ieiλckδkm,k∈1,q¯,m∈1,r-¯. And, consequently, (110)G~0y~j+=ie-iλbjδjll=1p,G~1y~j+=0,j∈1,p¯G~0y~k-=0,G~1y~k-=-ieiλckδkmm=1r-,k∈1,q¯. Hence(111)D~0λ=S1λ00000S2λ00000S3λ00:Cr-⊕Cm-︸Cq⊕Cν⊕Cr-⊕Cm-︸Cq→Cr-⊕Cm-⊕Cν︸Cp,D~1λ=0,0,0,S4λ,0:Cr-⊕Cm-︸Cq⊕Cν⊕Cr-⊕Cm-︸Cq→Cr-,where (112)S1λ=diag⁡ie-iλb1,…,ie-iλbr-(∈BCr-S2λ=diag⁡ie-iλbr-+1,…,ie-iλbq∈BCm-S3λ=diag⁡ie-iλbq+1,…,ie-iλbp∈BCνS4λ=diag⁡-ieiλc1,…,-ieiλcr-(∈BCr-. It follows from (87) and (66) that Gj=G~jU-1↾dom⁡Tmax,j∈{0,1}, where U is the unitary operator (65). This and (67) imply that Dj(λ)=D~j(λ)X, j∈{0,1}, and, consequently, (113)D0λ=G01YλG02YλG03Yλ=12S1λ00i2S1λ0012S2λ00i2S2λ00S3λ00D1λ=G1Yλ=12S4λ,0,0,-i2S4λ,0 (here G0j are entries of the block representation (82) of G0). Therefore by (83)–(85) the monodromy matrix B(λ)=Γ∞Y(λ) is (114)Bλ=B11λ0B13λ0B22λ0B31λ0B33λ:Cq⊕Cν⊕Cq→Cq⊕Cν⊕Cq, where(115)B11λ=diag⁡12S1λ+S4λ,12S2λ(116)B13λ=diag⁡i2S1λ-S4λ,i2S2λ(117)B22λ=S3λ,B31λ=diag⁡-i2S1λ-S4λ,-i22S2λ(118)B33λ=diag⁡12S1λ+S4λ,122S2λ,and the equalities (30)–(33) take the form(119)w1λ=B11λ0B13λ0-iB22λ-Iν000-Iq(120)w2λ=12B13λ0-B11λ0B22λ+Iν0Iq00(121)w3λ=B31λ0B33λ0-12B22λ+Iν0-Iq00(122)w4λ=12B33λ0-B31λ0-i2B22λ-Iν000-Iq.According to Theorem 10 equalities (34) and (35) with wj(λ) of the form (119)–(122) give a bijective correspondence σ^(s)=σ^τ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ^(s) of system (20). Combining of this statement with Proposition 14 implies that equality (105) with ωj(λ)=wj(λ)X∗,j∈1,4¯, and the equality (106) establish a bijective correspondence σ(s)=στ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ(·)∈SFn(I~). Moreover, the immediate calculation shows that ωj(λ) are of the form (101)–(104).In the casem-=0 spectral functions σ(·)∈SFn(I~) can be parameterized in a somewhat other form. Namely, the following theorem is valid.Theorem 17.
Assume thatp,q∈N∪{0}, q≤p, ν≔p-q, n=p+q, and(123)I~=0,b1,…,0,bp,-c1,0,…,-cq,0,where bj,cj∈(0,∞). Let(124)P1λ=diag⁡ie-iλb1,…,ie-iλbq,P2λ=diag⁡-ieiλc1,…,-ieiλcqP3λ=diag⁡ie-iλbq+1,…,ie-iλbp,λ∈C+ and let(125)Q1λ=00i2P2λ012P3λ-3Iν0-2Iq00∈BCq⊕Cν⊕Cq︸Cn(126)Q2λ=12002P2λ0i2P3λ+3Iν0i2Iq00∈BCq⊕Cν⊕Cq︸Cn(127)Q3λ=i2P1λ000123P3λ-Iν0002Iq∈BCq⊕Cν⊕Cq︸Cn(128)Q4λ=12-2P1λ000i23P3λ+Iν000i2Iq∈BCq⊕Cν⊕Cq︸Cn (in formulas (125)–(128) λ∈C+). Then the equalities(129)mKλ=KλQ1λ+Q3λ-1KλQ2λ+Q4λ,λ∈C+(130)σKs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡mKx+iydxestablish a bijective correspondence σ(s)=σK(s) between all contractive operator-functions K(·)∈S(Cn) and all spectral functions σ(·)∈SFn(I~).Proof.
Clearly, system of intervals (123) is system (77) with m-=0 and r-=q. Therefore the operator-functions ψj(λ) (see (97)–(100)) are ψ1(λ)=ψ3(λ)=P1(λ), ψ2(λ)=P2(λ),ψ4(λ)=P3(λ), and hence (131)Q1λ=iω1λ+ω3λ,Q2λ=iω2λ+ω4λQ3λ=iω1λ-ω3λ,Q4λ=iω2λ-ω4λ, where ωj(λ) are given by (101)–(104). Moreover, the subspace H in Theorem 16 coincides with Cn. Therefore R~H(Cn)=R~(Cn) and according to Proposition 8 the equalities(132)C0λ=iKλ+In,C1λ=Kλ-In,λ∈C+give a bijective correspondence between all operator-functions K(·)∈S(Cn) and all pairs τ={C0(·),C1(·)}∈R~(Cn). Substituting (132) into (105) one gets (133)mKλ=mτλ=iKλ+Inω1λ+Kλ-Inω3λ-1iKλ+Inω2λ+Kλ-Inω4λ=KλQ1λ+Q3λ-1KλQ2λ+Q4λ.Thus (105) admits the representation (129), which in view of Theorem 16 yields the required statement.


### 3.3. Particular Cases

In this subsection by using Theorems16 and 17 we obtain parametrization of spectral functions σ(·)∈SFn(I~) in certain particular cases.Corollary 18.
Assume thatp∈N,(134)I~=0,b1,…,0,bp(bj∈(0,∞)) and(135)Qλ=diag⁡e-iλb1,…,e-iλbp,λ∈C+.Then the equality(136)mKλ=i2Qλ-Kλ-1Qλ+Kλ,λ∈C+together with (130) gives a bijective correspondence σ(s)=σK(s) between all operator-functions K(·)∈S(Cp) and all spectral functions σ(·)∈SFp(I~).Proof.
Clearly, system of intervals (134) is system (123) with q=0, n=p, and hence the corresponding operator-functions Qj(λ) (see (125)–(128)) are(137)Q1λ=12iQλ-3Ip,Q2λ=i4iQλ+3Ip(138)Q3λ=123iQλ-Ip,Q4λ=i43iQλ+Ip.Let N(·)∈S(Cp) and let m(λ)=mN(λ) be given by (129) with N(λ) instead of K(λ). It follows from (137) and (138) that (139)NλQ1λ+Q3λ=i2Nλ+3IpQλ-KλNλQ2λ+Q4λ=-14Nλ+3IpQλ+Kλ,where(140)Kλ=-iNλ+3Ip-13Nλ+Ip=-i3Nλ+IpNλ+3Ip-1.Therefore m(λ)=mK(λ), where mK(λ) is given by (136) with K(λ) of the form (140).
LetX and J^ be operators in Cp⊕Cp defined by (141)X=183IpIpIp3Ip,J^=Ip00-Ip. Then X∗J^X=J^ and according to [14] equality (140) gives a bijection N(λ)→K(λ) of the set S(Cp) onto itself. This and Theorem 17 yield the desired statement.Corollary 19.
Assume thatp∈N, n=2p, and(142)I~=0,b1,…,0,bp,-c1,0,…,-cp,0(bj,cj∈(0,∞)). Let P1(λ) and P2(λ) be given by (124) and let Qj:C+→B(Cp⊕Cp) be the operator-functions defined by the block representations (143)Q1λ=0i2P2λ-2Ip0,Q2λ=i2P1λ002Ip,λ∈C+.Then statement of Theorem 17 holds with the equality(144)mKλ=-i2KλQ1λ+Q2λ-1KλQ1λ-Q2λ,λ∈C+instead of (129).Proof.
Since system of intervals (142) is system (123) with q=p, the operator-functions Qj(λ) (see (125)–(128)) are (145)Q1λ=Q1λ,Q2λ=-i2Q1λ,Q3λ=Q2λ,Q4λ=i2Q2λ.Therefore equality (129) admits the representation (144), which proves the required statement.Proposition 20.
Assume thatp∈N, n=2p,(146)I~=0,b1,…,0,bp,R-,…,R-︸ptimes(bj∈(0,∞)), and Q(λ) is given by (135). Then the equalities(147)mK1λ=i2Qλ-K1λ-1Qλ+K1λ(148)NKλ=K2λQλ-K1λ-1Qλ(149)mKλ=mK1λ0NKλi2Ip:Cp⊕Cp→Cp⊕Cp,λ∈C+ together with (130) establish a bijective correspondence σ(s)=σK(s) between all operator-functions K(·)∈S(Cp,Cn) with the block representation K(λ)=(K1(λ),K2(λ))⊤:Cp→Cp⊕Cp,λ∈C+, and all spectral functions σ(·)∈SFn(I~).Proof.
Clearly, system of intervals (146) is system (77) with r-=0 and q=m-=p. Therefore for this system Theorem 16 holds with ψ1(λ)=i2Q(λ), ψ2(λ)=0, ψ3(λ)=i/2Q(λ), and(150)ω1λ=12i2Qλ0iIp-iIp,ω2λ=122-2Qλ0IpIp(151)ω3λ=1212Qλ0-Ip-Ip,ω4λ=122i2Qλ0iIp-iIp, where ωj(λ)∈B(Cp⊕Cp). Moreover, the subspace H in Theorem 16 is H={0}⊕Cp and according to Proposition 8 the equalities(152)C0λ=12Ip-K1λ0K2λ+iIp,C1λ=i2IpiK1λ0-iK2λ-Ipgive a bijective correspondence between all operator-functions K(·)∈S(Cp,Cn) with the block representation K(λ)=(K1(λ),K2(λ))⊤:Cp→Cp⊕Cp and all pairs τ={C0(·),C1(·)}∈R~{0}⊕Cp(Cn).
It follows from (152) and (150), (151) that (153)C0λω1λ+C1λω3λ-1=2iQλ-K1λ0iK2λIp-1=12-iQλ-K1λ-10-K2λQλ-K1λ-1IpC0λω2λ+C1λω4λ=12-Qλ+K1λ0K2λiIp and substitution of these equalities into (105) gives (154)mKλ≔mτλ=mK1λ0NKλi2Ip, where mK1(λ) is of the form (147) and (155)NKλ=12K2λQλ-K1λ-1Qλ+K1λ+Ip=12K2λQλ-K1λ-1×Qλ+K1λ+Qλ-K1λ=K2λQλ-K1λ-1Qλ.Thus mK(λ) is of the form (149), which in view of Theorem 16 proves the required statement.Theorem 21.
Assume thatp∈N and I~ is a system of intervals given by(156)I~=R+,…,R+︸ptimes,R-,…,R-︸ptimes.Then the equalities(157)ΣFx=12πIpF∗xFxIp,x∈R,andσFs=∫0sΣFxdxgive a bijective correspondence between all operator-functions F∈SR(Cp) and all spectral functions σ(·)∈SFn(I~).Proof.
Clearly, system of intervals (156) is system (48) with r+=r-=0 and m+=m-=q=p. Therefore n=2p, ν(=p-q)=0, and (62) takes the form(158)X=12IpiIpIp-iIp,X∗=12IpIp-iIpiIp.Consider canonical system (20) associated with I~ (recall that J and H(t) in this system are given by (25) and (63), respectively). Clearly, J=0-IpIp0; that is, system (20) is a Hamiltonian system. Moreover, by (68) deficiency indices N± of this system are N+=N-=p. Therefore according to [15] the equalities(159)Γ0y=-y10,Γ1y=y00,y∈dom⁡Tmax define a boundary triplet Π={Cp,Γ0,Γ1} for Tmax (in (159) y0(t) and y1(t) are components of the representation y(t)={y0(t),y1(t)}(∈Cp⊕Cp), y∈dom⁡Tmax).
LetM(·) be the Weyl function of the triplet Π. Since Tmin is a densely defined operator, it follows from [16, Corollary 4.8] that the equality (160)m^τλ=MλC0λ-C1λMλ-1C0λ-12Ip-MλC0λ-C1λMλ-1C1λ12Ip-C0λ-C1λMλ-1C0λC0λ-C1λMλ-1C1λ(λ∈C+) together with (35) gives a bijective correspondence σ^(s)=σ^τ(s) between all pairs τ={C0(·),C1(·)}∈R~(Cp) and all spectral functions σ^(·) of system (20). Therefore by Proposition 14 the equality(161)mτλ=Xm^τλX∗ together with (106) gives a bijective correspondence σ(s)=στ(s) between all pairs τ={C0(·),C1(·)}∈R~(Cp) and all spectral functions σ(·)∈SFn(I~).
Let us calculateM(λ). Assume that y~k-=y~k-(·,λ), k∈1,p¯, λ∈C+, be vector-functions defined by the second equality in (55) and let yk-=yk-(t,λ)≔X∗y~k-(t,λ) (this means that yk-=Uy~k-, where U is the unitary operator (65)). It follows from (57) that {yk-}k=1p is a basis in Nλ(Tmin). Let Z(·,λ)(∈B(Cp,Cn)) be the operator solution of system (20) given by (162)Zt,λ=y1-t,λ,y2-t,λ,…,yp-t,λ,λ∈C+. Clearly,(163)Z0,λ=12IpiIp:Cp→Cp⊕Cp and the equality (Z(λ)x)(t)=Z(t,λ)x, x∈Cp, defines a bijective linear mapping Z(λ) from Cp onto Nλ(Tmin). It follows from (159) and (163) that (164)Γ0Zλ=-i2Ip,Γ1Zλ=12Ip,λ∈C+. Hence Γ1Z(λ)=iΓ0Z(λ),λ∈C+, and by definition (18) of the Weyl function one has(165)Mλ=iIp,λ∈C+.Moreover, according to Proposition 8 the equalities(166)C0λ=iKλ+Ip,C1λ=Kλ-Ip,λ∈C+ give a bijective correspondence between all operator-functions K∈S(Cp) and all pairs τ={C0(·),C1(·)}∈R~(Cp). Substituting (165) and (166) into (160) one obtains (167)m^Kλ≔m^τλ=12iKλ+Ip-Kλ-Kλ-iKλ-Ipand the immediate calculation with taking (158) and (161) into account gives(168)mKλ≔mτλ=i2Ip02KλIp.Thus equalities (168) and (130) give a bijective correspondence σ(s)=σK(s) between all operator-functions K∈S(Cp) and all spectral functions σ(·)∈SFn(I~). Next, in view of (168) one has (169)Im⁡mKx+iy=12IpK∗x+iyKx+iyIp.x∈R,y>0. Since for each finite interval Δ⊂R the function K(x+iy) is bounded on Δ×R, it follows from the Lebesgue theorem that σK(s) admits the representation σK(s)=∫[0,s)ΣF(x)dx, where ΣF(x) is given by the first equality in (157) with F(x)=limy→+0K(x+iy). Letting now σF(s)=σK(s), we obtain the required statement.Corollary 22.
Let eitherI~={R+,…,R+︸ptimes} or I~={R,…,R︸ptimes}. Then the set SFp(I~) consists of the unique spectral function σ(s)=1/2πsIp, s∈R.Proof.
First assume thatI~={R+,…,R+︸ptimes}. Let I~-={R-,…,R-︸ptimes}, let I~′=I~∪I~- (that is, I~′ is given by the right hand side of (156)), and let in (157) F(x)=0. Then by Theorem 21 the equality σ~(s)=diag⁡(σ(s),σ(s)) defines a spectral function σ~(·)∈SF2p(I~′). This implies that σ(·)∈SFp(I~)∩SFp(I~-).
Conversely, letσ1(·)∈SFp(I~). Then the equality σ~1(s)=diag⁡(σ1(s),σ(s)) defines a spectral function σ~1(·)∈SF2p(I~′). Clearly, σ~1(·)=σF(·) with F(x)=0 (see (157)) and hence σ1(s)=σ(s). This proves the required statement for I~={R+,…,R+}. In the case I~={R,…,R} the statement of the corollary is implied by the Plancherel theorem and the obvious inclusion SFp(R,…,R)⊂SFp(R+,…,R+).Remark 23.
The fact thatσ(s)=(1/2π)sIp is a spectral function of the classes SFp(R+,…,R+) and SFp(R,…,R) directly follows from the Plancherel theorem. Corollary 22 claims that there are no other spectral functions in these classes.


## 3.1. Definitions and Auxiliary Results

Let(36)I~=I1,I2,…,Inbe a system of intervals Ij=(aj,bj), -∞≤aj<bj≤∞, j∈1,n¯. Denote by L2(R,Cn;I~) the set of all functions φ(·)={φj(·)}j=1n∈L2(R,Cn) with φj(t)=0 a.e. on R∖Ij and by L˙2(R,Cn;I~) the set of all functions φ(·)={φj(·)}j=1n∈L2(R,Cn;I~) with the following property: if Ij∈I~ is an infinite interval, then there exists a finite interval (αj,βj)⊂Ij (depending on φ) such that φj(t)=0 a.e. on Ij∖(αj,βj). Since L2(R,Cn;I~) is unitarily equivalent to the Hilbert space L2(I1)⊕L2(I2)⊕⋯⊕L2(In), it follows that L2(R,Cn;I~) is a closed subspace in L2(R,Cn). Moreover, L˙2(R,Cn;I~) is a linear manifold dense in L2(R,Cn;I~).With each functionφ∈L˙2(R,Cn;I~) one associates the function φ^:R→Cn (the classical vector-valued Fourier transform of φ) given by(37)φ^s=∫Reitsφtdt,s∈R.Assume that σ(·):R→B(Cn) is an operator-valued distribution function such that for each φ∈L˙2(R,Cn;I~) the following Parseval identity holds:(38)φ^L2σ;Cn≔∫Rdσsφ^s,φ^s=∫Rφt2dt≕φL2R,Cn.Then for each φ∈L2(R,Cn;I~) the integral in (37) converges in the norm of L2(σ,Cn) and the equality Vσφ=φ^,φ∈L2(R,Cn;I~), defines an isometry Vσ from L2(R,Cn;I~) to L2(σ;Cn).Definition 11.
AB(Cn)-valued distribution function σ(·) such that (38) holds will be called a spectral function for the classical Fourier transform (37) (with respect to the system of intervals I~).In what follows the set of spectral functions in the sense of Definition11 will be denoted by SFn(I~)=SFn(I1,I2,…,In). Moreover, in the case when n=1 and the system I~ consists of a unique interval I=(a,b), we let SF(a,b)≔SF1(I~).Proposition 12.
LetI~ be system of intervals (36), let χIj(·) be the indicator of an interval Ij, let χI~(t)=diag⁡(χI1(t),χI2(t),…,χIn(t)), and let σ(·)∈SFn(I~). Then the inverse to (37) transform is(39)φt=∫Re-itsχI~tdσsφ^s,where the integral converges in the norm of L2(R,Cn). If in addition σ(·) is absolutely continuous and Σ(s)=σ′(s) is the density of σ(·), then Parseval identity (38) and the inverse transform (39) can be written as(40)∫RΣsφ^s,φ^sds=∫Rφt2dt,φ∈L˙2R,Cn;I~(41)φt=∫Re-itsχI~tΣsφ^sds.Proof.
SinceVσ is an isometry, to prove (39) it is sufficient to show that(42)Vσ∗gt=∫Re-itsχI~tdσsgs,g∈L˙2σ,Cn,where L˙2(σ,Cn) is the set of all functions g∈L2(σ,Cn) with compact support.
Letg∈L˙2(σ,Cn), let fg(t)={fg,j(t)}j=1n be given by (43)fgt=∫Re-itsχI~tdσsgs≔χI~t∫Re-itsΨsgsdμ (for Ψ and μ see assertions before (16)), and let f∗(t)={fj∗(t)}j=1n≔(Vσ∗g)(t)(∈L2(R,Cn;I~)). Let us show that for each finite interval Ij′⊂Ij(44)∫RχIj′tfg,jt¯dt=∫RχIj′tfj∗t¯dt,j∈1,n¯.Indeed, let αj(t)={δkjχIj′(t)}k=1n∈L2(R,Cn;I~). Then by using the Fubini theorem and (16) one gets (45)∫RχIj′tfg,jt¯dt=∫Rαjt,fgtdt=∫Rαjt,∫Re-itsΨsgsdμdt=∫R∫Rαjt,e-itsΨsgsdμdt=∫R∫Rαjt,e-itsΨsgsdtdμ=∫RΨs∫Reitsαjtdt,gsdμ=Vσαj,gL2σ;Cn=αj,Vσ∗gL2R,Cn;I~=∫Rαjt,f∗tdt=∫RχIj′tfj∗t¯dt,which proves (44). Since fg,j(t)=fj∗(t)=0 (a.e. on R∖Ij), it follows from (44) that fg(t)=f∗(t) (a.e. on R) and hence (42) holds.
The last statement of the proposition is obvious.Remark 13.
Ifn=1 and system I~ consists of a unique interval I=(a,b), then equality (37) defines the classical C-valued Fourier transform φ^ of a scalar function φ∈L˙2(R) with compact support belonging to (a,b) (we denote the set of such functions by L˙2(R;(a,b))). A spectral function σ(·)∈SF(a,b) for this transform is a scalar distribution function such that the Parseval identity (46)∫Rφ^s2dσs=∫Rφt2dt,φ∈L˙2R;a,bholds; moreover, the inverse Fourier transform is (47)φt=∫Re-itsχItφ^sdσs.Assume thatr±,m±∈N∪{0}, p≔r++m+,q≔r-+m-, n≔p+q, and I~ is a system of intervals given by(48)I~=0,b1,…,0,br+,R+,…,R+︸m+times︷p,-c1,0,…,-cr-,0,R-,…,R-︸m-times︷q(here bj,cj∈(0,∞)). With the system I~ one associates canonical differential system(49)J~y′=λH~ty,t∈R+,where(50)J~=idiag⁡Ip,-Iq,H~t=diag⁡h1+t,…,hp+t,h1-t,…,hq-t.In (50) hj+,j∈1,p¯, and hk-,k∈1,q¯, are continuous functions on R+ such that hj+(t)>0, hk-(t)>0, t∈R+, and(51)∫R+hj+tdt=bj,j∈1,r+¯;∫R+hj+tdt=+∞,j∈r++1,p¯(52)∫R+hk-tdt=ck,k∈1,r-¯;∫R+hk-tdt=+∞,k∈r-+1,q¯.One can easily verify that the matrizant Y~(·,λ) of system (49) is(53)Y~t,λ=diag⁡e-iλΦ1+t,…,e-iλΦp+t,eiλΦ1-t,…,eiλΦq-t,where(54)Φj+t=∫0,thj+udu,j∈1,p¯;Φk-t=∫0,thk-udu,k∈1,q¯.Assume that εj={δkj}k=1n∈Cn, j∈1,n¯, is the standard basis in Cn and let(55)y~j+t,λ=e-iλΦj+tεj,j∈1,p¯;y~k-t,λ=eiλΦk-tεp+k,k∈1,q¯.Since by (53) (y~j+(t,λ))⊤ and (y~k-(t,λ))⊤ are columns of the matrizant Y~(t,λ), the system {y~j+(·,λ)}j=1p∪{y~k-(·,λ)}k=1q is a basis in the space of all solutions of system (49). Moreover, the immediate checking shows that(56)y~j+·,λ∈LH~2R+,Cn,j∈1,r+¯;y~j+·,λ∉LH~2R+,Cn,j∈r++1,p¯(57)y~k-·,λ∈LH~2R+,Cn,k∈1,q¯for λ∈C+ and(58)y~j+·,λ∈LH~2R+,Cn,j∈1,p¯;y~k-·,λ∈LH~2R+,Cn,k∈1,r-¯(59)y~k-·,λ∉LH~2R+,Cn,k∈r-+1,q¯for λ∈C-. Therefore deficiency indices N~± of system (49) are(60)N~+=q+r+,N~-=p+r-.Now assume that I~ is system of intervals (48) with p≥q and let ν≔p-q, so that (61)p=q+ν=r-+m-+ν,n=p+q=2q+ν. Moreover, let X and X∗ be mutually adjoint unitary operators in Cn defined by(62)X=12Iq0iIq02Iν0Iq0-iIq,X∗=12Iq0Iq02Iν0-iIq0iIq.Together with canonical system (49), (50) we associate with system of intervals I~ canonical system (20), where J is of the form (25) and(63)Ht=X∗H~tX,t∈R+.Clearly, J~ admits the representation J~=diag⁡iIq,iIν,-iIq and the immediate checking shows that(64)J=X∗J~X,J~=XJX∗.In the following we denote by Tmax and T~max the maximal operators induced by systems (20) and (49), respectively. It follows from (63) and (64) that the equality(65)Uf~t=X∗f~t,f~∈LH~2R+,Cndefines a unitary operator U from LH~2(R+,Cn) onto LH2(R+,Cn) such that Udom⁡T~max=dom⁡Tmax and(66)JUy~t,Uz~t=J~y~t,z~t,Uy~,Uz~∞=y~,z~∞,y~,z~∈dom⁡T~max.Moreover, the matrizant Y(·,λ) of system (20) is(67)Yt,λ=X∗Y~t,λX,where Y~(t,λ) is the matrizant (53) of system (49). Observe also that systems (20) and (49) have the same deficiency indices and by (60) deficiency indices N± of the system (20) are(68)N+=q+r+,N-=p+r-.Proposition 14.
LetI~ be system of intervals (48). Then the set SFn(I~) coincides with the set of spectral functions of system (49), (50). If in addition p≥q, then the equality(69)σs=Xσ^sX∗gives a bijective correspondence between all spectral functions σ^(·) of system (20), (63) and all spectral functions σ(·)∈SFn(I~).Proof.
LetIj+=[0,bj), j∈1,r+¯, and Ij+=R+,j∈r++1,p¯. Moreover, let Ik-=-ck,0, k∈1,r-¯, and Ik-=R-, k∈r-+1,q¯. Clearly, Φj+(·) (Φk-(·)) is a strictly increasing absolutely continuous function mapping bijectively R+ onto Ij+ (resp. -Ik-={t∈R:-t∈Ik-}). With each vector-function(70)f·=f1+·,…,fp+·,f1-·,…,fq-·:R+→Cnwe associate the vector-function(71)φf·=φ1+·,…,φp+·,φ1-·,…,φq-·:R→Cngiven by (72)φj+u=fj+Φj+-1u,u∈Ij+0,u∉Ij+,φk-u=fk-Φk--1-u,u∈Ik-0,u∉Ik- (here j∈1,p¯ and k∈1,q¯). Since Φj+′(t)=hj+(t) and Φk-′(t)=hk-(t), it follows that (73)∫R+hj+tfj+t2dt=∫Ij+fj+Φj+-1u2du=∫Rφj+u2du,j∈1,p¯∫R+hk-tfk-t2dt=∫Ik-fk-Φk--1-u2du=∫Rφk-u2du,k∈1,q¯ and hence (74)∫R+H~tft,ftdt=Σj=1p∫R+hj+tfj+t2dt+Σk=1q∫R+hk-tfk-t2dt=Σj=1p∫Rφj+u2du+Σk=1q∫Rφk-u2du=∫Rφfu2du.Thus, the equality φ(t)=φf(t) gives a bijective correspondence between all functions f∈L˙H~2(R+,Cn) and all functions φ∈L˙2(R,Cn;I~). Moreover, fLH~2(R+,Cn)=φfL2(R,Cn;I~).
Next assume thatf∈L˙H~2(R+,Cn) and φf∈L˙2(R,Cn;I~) are vector-functions (70) and (71) and let (75)f^=f^1+,…,f^p+,f^1-,…,f^q-,φ^f=φ^1+,…,φ^p+,φ^1-,…,φ^q- be the corresponding Fourier transforms (23) and (37) (in (23) H(t) and Y(t,λ) should be replaced with H~(t) and Y~(t,λ), respectively). Then by (53) and (50) (76)f^j+s=∫R+eisΦj+thj+tfj+tdt=∫Ij+eisufj+Φj+-1udu=∫Reisuφj+udu=φ^j+sf^k-s=∫R+e-isΦk-thk-tfk-tdt=∫Ik-eisufk-Φk--1-udu=∫Reisuφk-udu=φ^k-sand, consequently, f^(s)=φ^f(s). This yields the first statement of the proposition.
Ifp≥q, then by (63) and (67) the equality (69) gives a bijective correspondence between all spectral functions σ^(·) of system (20), (63) and all spectral functions σ(·) of system (49), (50). This and the first statement of the proposition yield the second statement.Below unless otherwise stated we assume thatI~ is system of intervals (48) with m+=0 and p≥q. This means that(77)I~=0,b1,…,0,bp,-c1,0,…,-cr-,0,R-,…,R-︸m-times︷q,and according to (68) deficiency indices N± of the respective system (20) are(78)N+=q+p=n,N-=p+r-.Lemma 15.
Letεj={δkj}k=1n,j∈1,n¯, be the standard basis in Cn, let εj′=X∗εj and let(79)ujt=eΦj+t-bjεj′,j∈1,p¯;vkt=eΦk-t-ckεp+k′k∈1,r-¯,t∈R+,where Φj+(t) and Φk-(t) are given by (54). Then uj,vk∈dom⁡Tmax and the equalities(80)G0y=y,uj∞j=1p∈Cp,G1y=y,vk∞k=1r-∈Cr-,y∈dom⁡Tmaxdefine linear mappings G0:dom⁡Tmax→Cp and G1:dom⁡Tmax→Cr- such that the mapping G≔(G0,G1)⊤:dom⁡Tmax→Cp⊕Cr- is surjective and the identity(81)y,z∞=iG0y,G0zCp-iG1y,G1zCr-,y,z∈dom⁡Tmaxis valid. Moreover, if(82)G0=G01,G02,G03⊤:dom⁡Tmax→Cr-⊕Cm-⊕Cν︸Cpis the block representation of G0, then the equalities(83)Γ0∞=12G01+G1,G02⊤:dom⁡Tmax→Cr-⊕Cm-︸Cq(84)Γ^∞=G03:dom⁡Tmax→Cν(85)Γ1∞-i2G01-G1,-i2G02⊤:dom⁡Tmax→Cr-⊕Cm-︸Cqtogether with (26) define the operator Γ∞ such that (27) and (28) hold.Proof.
Let(86)u~jt=eΦj+t-bjεj,j∈1,p¯;v~kt=eΦk-t-ckεp+k,k∈1,r-¯.Then by (55) u~j(t)=e-bjy~j+(t,i), v~k(t)=e-cky~k-(t,-i), and in view of (56) and (58) u~j,v~k∈dom⁡T~max. Since uj(t)=X∗u~j(t) and vk(t)=X∗v~k(t), it follows that(87)uj=Uu~j,j∈1,p¯;vk=Uv~k,k∈1,r-¯,where U is the unitary operator (65). Hence uj,vk∈dom⁡Tmax.
Clearly,(88)J~εj,εk=iδjk,j,k∈1,p¯0,j∈1,p¯,k∈p+1,n,¯-iδjk,j,k∈p+1,n¯.This and (66), (87) yield (89)Jujt,ult=J~u~jt,u~lt=ieΦj+t-bjeΦl+t-blδjl,j,l∈1,p¯Jujt,vkt=J~u~jt,v~kt=0,j∈1,p¯,k∈1,r-¯Jvkt,vmt=J~v~kt,v~mt=-ieΦk-t-ckeΦm-t-cmδkm,k,m∈1,r-¯. Since by (51) and (52) Φj+(∞)=bj, j∈1,p¯, and Φk-(∞)=ck, k∈1,r-¯, it follows that(90)uj,ul∞=iδjl,j,l∈1,p¯;uj,vk∞=0,j∈1,p¯,k∈1,r-¯(91)vk,vm∞=-iδkm,k,m∈1,r-¯.Next, denote by D0 the kernel of the form [·,·]∞. According to [13, Proposition 5.5] the indices of inertia ν∞± of the form [·,·]∞ are ν∞+=N+-q,ν∞-=N--p and (78) yields ν∞+=p,ν∞-=r-. Therefore by (90) and (91) system {uj}j=1p∪{vk}k=1r- forms a basis in dom⁡Tmax/D0 and hence each y∈dom⁡Tmax admits a unique representation (92)y=y0+Σj=1pαjyuj+Σk=1r-βkyvk, where y0∈D0 and αjy,βky∈C. Let G0:dom⁡Tmax→Cp and G1:dom⁡Tmax→Cr- be the operators given by(93)G0y=iαjyj=1p∈Cp,G1y=-iβkyk=1r-∈Cr-.Then by (90) and (91) for any y,z∈dom⁡Tmax one has (94)y,z∞=iΣj=1pαjyαjz¯-iΣk=1r-βkyβkz¯=iG0y,G0zCp-iG1y,G1zCr-.Moreover, by (93) the operator G=(G0,G1)⊤ is surjective. Note also that by (90) and (91) [y,uj]∞=iαjy, [y,vk]∞=-iβky, and hence (93) admits the representation (80). This proves the first statement of the lemma.
Now assume thatΓ0∞,Γ^∞ and Γ1∞ are the operators (83)–(85), Γ∞ is given by (26), and P is the orthoprojector in Cq onto Cm-. Then (95)PΓ1∞=-i2G02=-i2PΓ0∞, which in view of surjectivity of G yields (27). Finally, for any y,z∈dom⁡Tmax one has (96)Γ0∞y,Γ1∞z-Γ1∞y,Γ0∞z+iΓ^∞y,Γ^∞z=i2G01y+G1y,G01z-G1z+i2G02y,G02z+i2G01y-G1y,G01z+G1z+i2G02y,G02z+iG03y,G03z=iG01y,G01z-iG1y,G1z+iG02y,G02z+iG03y,G03z=iG0y,G0z-iG1y,G1z=y,z∞.This proves (28).


## 3.2. Parametrization of Spectral Functions

A parametrization of all spectral functionsσ(·)∈SFn(I~) is given by the following theorem.Theorem 16.
Assume thatp,r-,m-∈N∪{0}, q≔r-+m-≤p, ν≔p-q, n≔p+q, and I~ is a system of intervals (77). Let H≔Cr-⊕Cν⊕Cq, let ψj:C+→B(Cq),j∈1,3¯ and ψ4:C+→B(Cν) be operator (matrix) functions given by(97)ψ1λ=diag⁡ie-iλb1,…,ie-iλbr-,i2e-iλbr-+1,…,i2e-iλbq(98)ψ2λ=diag⁡-ieiλc1,…,-ieiλcr-,0,…,0︸m-times(99)ψ3λ=diag⁡ie-iλb1,…,ie-iλbr-,i2e-iλbr-+1,…,i2e-iλbq(100)ψ4λ=diag⁡ie-iλbq+1,…,ie-iλbp,and let(101)ω1λ=12ψ1λ0ψ2λ0-i2ψ4λ-Iν0iIq0-iIq∈BCq⊕Cν⊕Cq︸Cn(102)ω2λ=122iψ1λ0-iψ2λ02ψ4λ+Iν0Iq0Iq∈BCq⊕Cν⊕Cq︸Cn(103)ω3λ=12-iψ3λ0iψ2λ0-12ψ4λ+Iν0-Iq0-Iq∈BCq⊕Cν⊕Cq︸Cn(104)ω4λ=122ψ3λ0ψ2λ0-i2ψ4λ-Iν0iIq0-iIq∈BCq⊕Cν⊕Cq︸Cn(in formulas (97)–(104) λ∈C+). Then the equalities(105)mτλ=C0λω1λ+C1λω3λ-1C0λω2λ+C1λω4λ,λ∈C+(106)στs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡mτx+iydxestablish a bijective correspondence σ(s)=στ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ(·)∈SFn(I~).Proof.
First by using Theorem10 we parameterize spectral functions of system (20) with J of the form (25) and H(t) defined by (63) and (50).
LetΓ∞ be the operator defined by (83)–(85) and (26) and let Y(·,λ) be the matrizant of system (20). Since N+=n (see (78)) and by Lemma 15Γ∞ satisfies (27) and (28), the equality B(λ)=Γ∞Y(λ), λ∈C+, defines the monodromy matrix B(λ). To calculate B(λ) we first calculate the operators D0(λ)=G0Y(λ)∈B(Cn,Cp) and D1(λ)=G1Y(λ)∈B(Cn,Cr-),λ∈C+ (here G0 and G1 are operators (80)). Let u~j,v~k∈dom⁡T~max be given by (86) and let G~0:dom⁡T~max→Cp and G~1:dom⁡T~max→Cr- be operators defined by (107)G~0y~=y~,u~j∞j=1p∈Cp,G~1y=y~,v~k∞k=1r-∈Cr-,y~∈dom⁡T~max. Moreover, let Y~(·,λ) be the matrizant (53) of system (49), (50), let Y~(λ) be the respective operator (22), and let D~0(λ)=G~0Y~(λ)∈B(Cn,Cp), D~1(λ)=G~1Y~(λ)∈B(Cn,Cr-),λ∈C+. Since (y~j+)⊤ and (y~k-)⊤ (see (55)) are columns of the matrizant (53), it follows that (108)D~mλ=G~my~1+⊤,…,G~my~p+⊤,G~my~1-⊤,…,G~my~q-⊤,m∈0,1. Moreover, by (88) (109)y~j+,u~l∞=limt→+∞⁡J~y~j+t,λ,u~lt=limt→+∞⁡ie-iλΦj+teΦl+t-blδjl=ie-iλbjδjl,j,l∈1,p¯y~j+,v~k∞=limt→+∞⁡J~y~j+t,λ,v~kt=0,j∈1,p¯,k∈1,r-¯y~k-,u~j∞=limt→+∞⁡J~y~k-t,λ,u~jt=0,k∈1,q¯,j∈1,p¯y~k-,v~m∞=limt→+∞⁡J~y~k-t,λ,v~mt=limt→+∞⁡-ieiλΦk-teΦm-t-cmδkm=-ieiλckδkm,k∈1,q¯,m∈1,r-¯. And, consequently, (110)G~0y~j+=ie-iλbjδjll=1p,G~1y~j+=0,j∈1,p¯G~0y~k-=0,G~1y~k-=-ieiλckδkmm=1r-,k∈1,q¯. Hence(111)D~0λ=S1λ00000S2λ00000S3λ00:Cr-⊕Cm-︸Cq⊕Cν⊕Cr-⊕Cm-︸Cq→Cr-⊕Cm-⊕Cν︸Cp,D~1λ=0,0,0,S4λ,0:Cr-⊕Cm-︸Cq⊕Cν⊕Cr-⊕Cm-︸Cq→Cr-,where (112)S1λ=diag⁡ie-iλb1,…,ie-iλbr-(∈BCr-S2λ=diag⁡ie-iλbr-+1,…,ie-iλbq∈BCm-S3λ=diag⁡ie-iλbq+1,…,ie-iλbp∈BCνS4λ=diag⁡-ieiλc1,…,-ieiλcr-(∈BCr-. It follows from (87) and (66) that Gj=G~jU-1↾dom⁡Tmax,j∈{0,1}, where U is the unitary operator (65). This and (67) imply that Dj(λ)=D~j(λ)X, j∈{0,1}, and, consequently, (113)D0λ=G01YλG02YλG03Yλ=12S1λ00i2S1λ0012S2λ00i2S2λ00S3λ00D1λ=G1Yλ=12S4λ,0,0,-i2S4λ,0 (here G0j are entries of the block representation (82) of G0). Therefore by (83)–(85) the monodromy matrix B(λ)=Γ∞Y(λ) is (114)Bλ=B11λ0B13λ0B22λ0B31λ0B33λ:Cq⊕Cν⊕Cq→Cq⊕Cν⊕Cq, where(115)B11λ=diag⁡12S1λ+S4λ,12S2λ(116)B13λ=diag⁡i2S1λ-S4λ,i2S2λ(117)B22λ=S3λ,B31λ=diag⁡-i2S1λ-S4λ,-i22S2λ(118)B33λ=diag⁡12S1λ+S4λ,122S2λ,and the equalities (30)–(33) take the form(119)w1λ=B11λ0B13λ0-iB22λ-Iν000-Iq(120)w2λ=12B13λ0-B11λ0B22λ+Iν0Iq00(121)w3λ=B31λ0B33λ0-12B22λ+Iν0-Iq00(122)w4λ=12B33λ0-B31λ0-i2B22λ-Iν000-Iq.According to Theorem 10 equalities (34) and (35) with wj(λ) of the form (119)–(122) give a bijective correspondence σ^(s)=σ^τ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ^(s) of system (20). Combining of this statement with Proposition 14 implies that equality (105) with ωj(λ)=wj(λ)X∗,j∈1,4¯, and the equality (106) establish a bijective correspondence σ(s)=στ(s) between all pairs τ={C0(·),C1(·)}∈R~H(Cn) and all spectral functions σ(·)∈SFn(I~). Moreover, the immediate calculation shows that ωj(λ) are of the form (101)–(104).In the casem-=0 spectral functions σ(·)∈SFn(I~) can be parameterized in a somewhat other form. Namely, the following theorem is valid.Theorem 17.
Assume thatp,q∈N∪{0}, q≤p, ν≔p-q, n=p+q, and(123)I~=0,b1,…,0,bp,-c1,0,…,-cq,0,where bj,cj∈(0,∞). Let(124)P1λ=diag⁡ie-iλb1,…,ie-iλbq,P2λ=diag⁡-ieiλc1,…,-ieiλcqP3λ=diag⁡ie-iλbq+1,…,ie-iλbp,λ∈C+ and let(125)Q1λ=00i2P2λ012P3λ-3Iν0-2Iq00∈BCq⊕Cν⊕Cq︸Cn(126)Q2λ=12002P2λ0i2P3λ+3Iν0i2Iq00∈BCq⊕Cν⊕Cq︸Cn(127)Q3λ=i2P1λ000123P3λ-Iν0002Iq∈BCq⊕Cν⊕Cq︸Cn(128)Q4λ=12-2P1λ000i23P3λ+Iν000i2Iq∈BCq⊕Cν⊕Cq︸Cn (in formulas (125)–(128) λ∈C+). Then the equalities(129)mKλ=KλQ1λ+Q3λ-1KλQ2λ+Q4λ,λ∈C+(130)σKs=limδ→+0⁡limy→+0⁡1π∫-δs-δIm⁡mKx+iydxestablish a bijective correspondence σ(s)=σK(s) between all contractive operator-functions K(·)∈S(Cn) and all spectral functions σ(·)∈SFn(I~).Proof.
Clearly, system of intervals (123) is system (77) with m-=0 and r-=q. Therefore the operator-functions ψj(λ) (see (97)–(100)) are ψ1(λ)=ψ3(λ)=P1(λ), ψ2(λ)=P2(λ),ψ4(λ)=P3(λ), and hence (131)Q1λ=iω1λ+ω3λ,Q2λ=iω2λ+ω4λQ3λ=iω1λ-ω3λ,Q4λ=iω2λ-ω4λ, where ωj(λ) are given by (101)–(104). Moreover, the subspace H in Theorem 16 coincides with Cn. Therefore R~H(Cn)=R~(Cn) and according to Proposition 8 the equalities(132)C0λ=iKλ+In,C1λ=Kλ-In,λ∈C+give a bijective correspondence between all operator-functions K(·)∈S(Cn) and all pairs τ={C0(·),C1(·)}∈R~(Cn). Substituting (132) into (105) one gets (133)mKλ=mτλ=iKλ+Inω1λ+Kλ-Inω3λ-1iKλ+Inω2λ+Kλ-Inω4λ=KλQ1λ+Q3λ-1KλQ2λ+Q4λ.Thus (105) admits the representation (129), which in view of Theorem 16 yields the required statement.


## 3.3. Particular Cases

In this subsection by using Theorems16 and 17 we obtain parametrization of spectral functions σ(·)∈SFn(I~) in certain particular cases.Corollary 18.
Assume thatp∈N,(134)I~=0,b1,…,0,bp(bj∈(0,∞)) and(135)Qλ=diag⁡e-iλb1,…,e-iλbp,λ∈C+.Then the equality(136)mKλ=i2Qλ-Kλ-1Qλ+Kλ,λ∈C+together with (130) gives a bijective correspondence σ(s)=σK(s) between all operator-functions K(·)∈S(Cp) and all spectral functions σ(·)∈SFp(I~).Proof.
Clearly, system of intervals (134) is system (123) with q=0, n=p, and hence the corresponding operator-functions Qj(λ) (see (125)–(128)) are(137)Q1λ=12iQλ-3Ip,Q2λ=i4iQλ+3Ip(138)Q3λ=123iQλ-Ip,Q4λ=i43iQλ+Ip.Let N(·)∈S(Cp) and let m(λ)=mN(λ) be given by (129) with N(λ) instead of K(λ). It follows from (137) and (138) that (139)NλQ1λ+Q3λ=i2Nλ+3IpQλ-KλNλQ2λ+Q4λ=-14Nλ+3IpQλ+Kλ,where(140)Kλ=-iNλ+3Ip-13Nλ+Ip=-i3Nλ+IpNλ+3Ip-1.Therefore m(λ)=mK(λ), where mK(λ) is given by (136) with K(λ) of the form (140).
LetX and J^ be operators in Cp⊕Cp defined by (141)X=183IpIpIp3Ip,J^=Ip00-Ip. Then X∗J^X=J^ and according to [14] equality (140) gives a bijection N(λ)→K(λ) of the set S(Cp) onto itself. This and Theorem 17 yield the desired statement.Corollary 19.
Assume thatp∈N, n=2p, and(142)I~=0,b1,…,0,bp,-c1,0,…,-cp,0(bj,cj∈(0,∞)). Let P1(λ) and P2(λ) be given by (124) and let Qj:C+→B(Cp⊕Cp) be the operator-functions defined by the block representations (143)Q1λ=0i2P2λ-2Ip0,Q2λ=i2P1λ002Ip,λ∈C+.Then statement of Theorem 17 holds with the equality(144)mKλ=-i2KλQ1λ+Q2λ-1KλQ1λ-Q2λ,λ∈C+instead of (129).Proof.
Since system of intervals (142) is system (123) with q=p, the operator-functions Qj(λ) (see (125)–(128)) are (145)Q1λ=Q1λ,Q2λ=-i2Q1λ,Q3λ=Q2λ,Q4λ=i2Q2λ.Therefore equality (129) admits the representation (144), which proves the required statement.Proposition 20.
Assume thatp∈N, n=2p,(146)I~=0,b1,…,0,bp,R-,…,R-︸ptimes(bj∈(0,∞)), and Q(λ) is given by (135). Then the equalities(147)mK1λ=i2Qλ-K1λ-1Qλ+K1λ(148)NKλ=K2λQλ-K1λ-1Qλ(149)mKλ=mK1λ0NKλi2Ip:Cp⊕Cp→Cp⊕Cp,λ∈C+ together with (130) establish a bijective correspondence σ(s)=σK(s) between all operator-functions K(·)∈S(Cp,Cn) with the block representation K(λ)=(K1(λ),K2(λ))⊤:Cp→Cp⊕Cp,λ∈C+, and all spectral functions σ(·)∈SFn(I~).Proof.
Clearly, system of intervals (146) is system (77) with r-=0 and q=m-=p. Therefore for this system Theorem 16 holds with ψ1(λ)=i2Q(λ), ψ2(λ)=0, ψ3(λ)=i/2Q(λ), and(150)ω1λ=12i2Qλ0iIp-iIp,ω2λ=122-2Qλ0IpIp(151)ω3λ=1212Qλ0-Ip-Ip,ω4λ=122i2Qλ0iIp-iIp, where ωj(λ)∈B(Cp⊕Cp). Moreover, the subspace H in Theorem 16 is H={0}⊕Cp and according to Proposition 8 the equalities(152)C0λ=12Ip-K1λ0K2λ+iIp,C1λ=i2IpiK1λ0-iK2λ-Ipgive a bijective correspondence between all operator-functions K(·)∈S(Cp,Cn) with the block representation K(λ)=(K1(λ),K2(λ))⊤:Cp→Cp⊕Cp and all pairs τ={C0(·),C1(·)}∈R~{0}⊕Cp(Cn).
It follows from (152) and (150), (151) that (153)C0λω1λ+C1λω3λ-1=2iQλ-K1λ0iK2λIp-1=12-iQλ-K1λ-10-K2λQλ-K1λ-1IpC0λω2λ+C1λω4λ=12-Qλ+K1λ0K2λiIp and substitution of these equalities into (105) gives (154)mKλ≔mτλ=mK1λ0NKλi2Ip, where mK1(λ) is of the form (147) and (155)NKλ=12K2λQλ-K1λ-1Qλ+K1λ+Ip=12K2λQλ-K1λ-1×Qλ+K1λ+Qλ-K1λ=K2λQλ-K1λ-1Qλ.Thus mK(λ) is of the form (149), which in view of Theorem 16 proves the required statement.Theorem 21.
Assume thatp∈N and I~ is a system of intervals given by(156)I~=R+,…,R+︸ptimes,R-,…,R-︸ptimes.Then the equalities(157)ΣFx=12πIpF∗xFxIp,x∈R,andσFs=∫0sΣFxdxgive a bijective correspondence between all operator-functions F∈SR(Cp) and all spectral functions σ(·)∈SFn(I~).Proof.
Clearly, system of intervals (156) is system (48) with r+=r-=0 and m+=m-=q=p. Therefore n=2p, ν(=p-q)=0, and (62) takes the form(158)X=12IpiIpIp-iIp,X∗=12IpIp-iIpiIp.Consider canonical system (20) associated with I~ (recall that J and H(t) in this system are given by (25) and (63), respectively). Clearly, J=0-IpIp0; that is, system (20) is a Hamiltonian system. Moreover, by (68) deficiency indices N± of this system are N+=N-=p. Therefore according to [15] the equalities(159)Γ0y=-y10,Γ1y=y00,y∈dom⁡Tmax define a boundary triplet Π={Cp,Γ0,Γ1} for Tmax (in (159) y0(t) and y1(t) are components of the representation y(t)={y0(t),y1(t)}(∈Cp⊕Cp), y∈dom⁡Tmax).
LetM(·) be the Weyl function of the triplet Π. Since Tmin is a densely defined operator, it follows from [16, Corollary 4.8] that the equality (160)m^τλ=MλC0λ-C1λMλ-1C0λ-12Ip-MλC0λ-C1λMλ-1C1λ12Ip-C0λ-C1λMλ-1C0λC0λ-C1λMλ-1C1λ(λ∈C+) together with (35) gives a bijective correspondence σ^(s)=σ^τ(s) between all pairs τ={C0(·),C1(·)}∈R~(Cp) and all spectral functions σ^(·) of system (20). Therefore by Proposition 14 the equality(161)mτλ=Xm^τλX∗ together with (106) gives a bijective correspondence σ(s)=στ(s) between all pairs τ={C0(·),C1(·)}∈R~(Cp) and all spectral functions σ(·)∈SFn(I~).
Let us calculateM(λ). Assume that y~k-=y~k-(·,λ), k∈1,p¯, λ∈C+, be vector-functions defined by the second equality in (55) and let yk-=yk-(t,λ)≔X∗y~k-(t,λ) (this means that yk-=Uy~k-, where U is the unitary operator (65)). It follows from (57) that {yk-}k=1p is a basis in Nλ(Tmin). Let Z(·,λ)(∈B(Cp,Cn)) be the operator solution of system (20) given by (162)Zt,λ=y1-t,λ,y2-t,λ,…,yp-t,λ,λ∈C+. Clearly,(163)Z0,λ=12IpiIp:Cp→Cp⊕Cp and the equality (Z(λ)x)(t)=Z(t,λ)x, x∈Cp, defines a bijective linear mapping Z(λ) from Cp onto Nλ(Tmin). It follows from (159) and (163) that (164)Γ0Zλ=-i2Ip,Γ1Zλ=12Ip,λ∈C+. Hence Γ1Z(λ)=iΓ0Z(λ),λ∈C+, and by definition (18) of the Weyl function one has(165)Mλ=iIp,λ∈C+.Moreover, according to Proposition 8 the equalities(166)C0λ=iKλ+Ip,C1λ=Kλ-Ip,λ∈C+ give a bijective correspondence between all operator-functions K∈S(Cp) and all pairs τ={C0(·),C1(·)}∈R~(Cp). Substituting (165) and (166) into (160) one obtains (167)m^Kλ≔m^τλ=12iKλ+Ip-Kλ-Kλ-iKλ-Ipand the immediate calculation with taking (158) and (161) into account gives(168)mKλ≔mτλ=i2Ip02KλIp.Thus equalities (168) and (130) give a bijective correspondence σ(s)=σK(s) between all operator-functions K∈S(Cp) and all spectral functions σ(·)∈SFn(I~). Next, in view of (168) one has (169)Im⁡mKx+iy=12IpK∗x+iyKx+iyIp.x∈R,y>0. Since for each finite interval Δ⊂R the function K(x+iy) is bounded on Δ×R, it follows from the Lebesgue theorem that σK(s) admits the representation σK(s)=∫[0,s)ΣF(x)dx, where ΣF(x) is given by the first equality in (157) with F(x)=limy→+0K(x+iy). Letting now σF(s)=σK(s), we obtain the required statement.Corollary 22.
Let eitherI~={R+,…,R+︸ptimes} or I~={R,…,R︸ptimes}. Then the set SFp(I~) consists of the unique spectral function σ(s)=1/2πsIp, s∈R.Proof.
First assume thatI~={R+,…,R+︸ptimes}. Let I~-={R-,…,R-︸ptimes}, let I~′=I~∪I~- (that is, I~′ is given by the right hand side of (156)), and let in (157) F(x)=0. Then by Theorem 21 the equality σ~(s)=diag⁡(σ(s),σ(s)) defines a spectral function σ~(·)∈SF2p(I~′). This implies that σ(·)∈SFp(I~)∩SFp(I~-).
Conversely, letσ1(·)∈SFp(I~). Then the equality σ~1(s)=diag⁡(σ1(s),σ(s)) defines a spectral function σ~1(·)∈SF2p(I~′). Clearly, σ~1(·)=σF(·) with F(x)=0 (see (157)) and hence σ1(s)=σ(s). This proves the required statement for I~={R+,…,R+}. In the case I~={R,…,R} the statement of the corollary is implied by the Plancherel theorem and the obvious inclusion SFp(R,…,R)⊂SFp(R+,…,R+).Remark 23.
The fact thatσ(s)=(1/2π)sIp is a spectral function of the classes SFp(R+,…,R+) and SFp(R,…,R) directly follows from the Plancherel theorem. Corollary 22 claims that there are no other spectral functions in these classes.



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