# On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

Authors: M. De la Sen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.


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## Body

## 1. Introduction

Compact operators in infinite-dimensional separable Hilbert spaces are of relevance in the study of certain relevant applied problems in control theory and signal theory, [1]. A natural property of such operators is that they can be represented with expansions using two orthogonal or orthonormal reciprocal bases of the separable Hilbert space. If the bases are orthonormal then both of them coincide so that this basis is autoreciprocal and then the formal study is facilitated [1, 2]. Many of the involved operators in mapping map an input space into an output space in the above problems are in addition self-adjoint. Another property of such operators is that they admit truncations using a finite number of the members of the orthonormal basis so that the truncated operators are also compact in a natural way, [1, 2]. The truncated operator describes a natural orthogonal projection of the involved vectors of the Hilbert space into a finite-dimensional space whose dimension is deceased as the number of members of the basis used for its representation decreases. On the other hand, important attention is being devoted to many aspects of fixed point theory in metric, Banach, and more general spaces including the study of mappings being contractive, nonexpansive, asymptotically contractive, asymptotically nonexpansive, quasi-nonexpansive, Kannan and Meir-Keeler and cyclic-type contractions, and so forth. Also, it has been studied the relevance of the theory in properties in both general theory and applications such as the existence and uniqueness of solutions in differential, difference, and hybrid equations as well as in continuous-time, discrete-time, and hybrid dynamic systems, stability theory in the above problems [3–7], the existence/uniqueness of fixed points and best proximity points, and the boundedness of iterated sequences being constructed through the maps and the convergence of such iterated calculations to limit points. See, for instance, [3–6, 8–15] and the references therein. The investigation of existence and uniqueness of common fixed points and best proximity points for several mappings and related properties is also important [10–12]. The study of fixed and best proximity points has also inherent study of convergence of sequences to such points. Other studies of properties of convergence of sequences and operator sequences have been described in different problems as, for instance, the research on approximating operators and approximation theorems that of sigma convergence of double sequences or that of lamda-statistical convergence and summability. See, for instance, [13–17] and the references therein.This paper is devoted to the investigation of self-adjoint compact operators in separable Hilbert spaces, their finite-dimensional truncated counterparts, and the relations in-between the corresponding properties for the norms of the mutual errors end the errors in-between the corresponding fixed points and their respective convergence properties when iterated calculations through the operators are performed. Some examples of interest in signal theory and control theory are also given. The operators and the iterated sequences constructed through them are studied by using the expansions of the operators and their finite dimensional truncated versions by using a numerable orthonormal basis of the involved Hilbert space.


## 2. Preliminaries and Main Results

The following result includes some properties related to the approximations ofx∈V and x∈H through orthonormal systems of different dimensions, complete orthonormal systems in H, and orthonormal basis, that is, a maximal orthonormal system; that is, it is not a proper subset of any orthonormal system of H, where V and H are an inner product space and a Hilbert space, respectively. Note that in the case where H is separable, a complete orthonormal system is always an orthonormal basis and vice versa.Lemma 1.
LetV be an inner product space of inner product 〈·,·〉:H×H→C (or R) endowed with a norm ∥·∥:V→R0+ defined by ∥x∥=〈x,x〉1/2 for any x∈H, where R0+={z∈R:z≥0}, let {en}n=1N and {an}n=1N be a finite orthonormal system in V and a given finite or numerable sequence of scalars, respectively, and let M and N be given integers fulfilling 1≤M≤N≤∞. If N=∞ then {en}n=1N is, in addition, assumed to be numerable. Then, the following properties hold for any x∈H.(i)
∥x-∑n=1Nanen∥2=∥x-∑n=M+1Nanen∥2-∑n=1M|〈x,en〉|2+∑n=1M|an-〈x,en〉|2.(ii)
∥x-∑n=1Manen∥2=∥x+∑n=M+1Nanen∥2-∑n=1N|〈x,en〉|2+∑n=1N|an-〈x,en〉|2.(iii)
∥x-∑n=1Nanen∥2-∥x-∑n=1Manen∥2=∑n=M+1N(|〈x,en〉|2+|〈x,en〉-an|2).(iv)
∥∑n=ijanen∥2=∑n=ij|an|2 any integers i,j(≥i)∈N-={1,2,…,N}.(v)
IfV=H is a finite-dimensional Hilbert space of dimension N and an=〈x,en〉, for all n∈N- and N=M, then
(1)∥x-∑n=1N〈x,en〉en∥2+∑n=1N|〈x,en〉|2=∥x∥2≥∑n=1N|〈x,en〉|2≥∑n=1M|〈x,en〉|2.(vi)
IfV=H is a finite-dimensional Hilbert space of dimension N and an=〈x,en〉, for n∈N-, then
(2)∥x-∑n=1Manen∥2≤2∑n=1N|an|2-3∑n=1M|an|2.(vii)
IfV=H is a separable infinite-dimensional Hilbert space and an=〈x,en〉, for n∈N, then
(3)∥x-∑n=1Manen∥2≤(∑n=1∞|an|2)≤∥x∥2.If, in addition, ∥x-∑n=1Manen∥2<+∞, then an→0asn(∈N)→∞. If, furthermore, there is some integer α≥M such that the real sequence {|an|}n≥α converges to zero exponentially according to |an|≤ρn≤ρ<1, for n∈N, then ∥x-∑n=1Manen∥2≤C(α)+ρ/(1-ρ) for any given x∈H with ρ∈(0,1) being some real constant and C(α) being a bounded constant dependent on α satisfying C(M)=0.Proof.
Properties (i)-(ii) follow from the best approximation lemma since(4)∥x-∑n=1Nanen∥2=∥(x-∑n=M+1Manen)-∑n=1Manen∥2=∥x-∑n=M+1Nanen∥2-∑n=1M|〈x,en〉|2+∑n=1M|an-〈x,en〉|2,∥x-∑n=1Manen∥2=∥(x+∑n=M+1Manen)-∑n=1Nanen∥2=∥x-∑n=M+1Nanen∥2-∑n=1N|〈x,en〉|2+∑n=1N|an-〈x,en〉|2.
Property (iii) is a direct consequence of subtracting both sides of the relations in Properties (i)-(ii). Property (iv) is Pythagoras theorem in inner product spaces. Property (v) (Bessel’s inequality) follows directly from Property (i) with the orthonormal system {en}n=1N in the Hilbert space H being a basis of H. Property (vi) follows from Properties (ii)–(iii) with an=〈x,en〉; n∈N- and the orthonormal system {en}n=1N in H being an orthonormal basis of H since one gets from Property (i)
(5)∥x-∑n=1Nanen∥2=∥x-∑n=M+1Nanen∥2-∑n=1M|〈x,en〉|2+∑n=1M|an-〈x,en〉|2=∥x-∑n=M+1Nanen∥2-∑n=1M|an|2=0⟹∥x-∑n=M+1Nanen∥2=∑n=1M|an|2
and from (5), Property (ii), and an=〈x,en〉, n∈N-(6)∥x-∑n=1Manen∥2=∥x+∑n=M+1Nanen-∑n=1Nanen∥2-∑n=1M|an|2≤2∥x-∑n=1Nanen∥2+2∥∑n=M+1Nanen∥2-∑n=1M|an|2=2∥∑n=M+1Nanen∥2-∑n=1M|an|2=2∑n=M+1N|an|2-∑n=1M|an|2=2∑n=1N|an|2-3∑n=1M|an|2.
Hence, Property (vi). Property (vii) follows from the assumption that the infinite-dimensional Hilbert space is separable and Property (vi) leads to
(7)∥x-∑n=1Manxn∥2=∑n=M+1∞|an|2≤∑n=1∞|an|2≤∥x∥2<+∞⟹(an⟶0asn(∈N)⟶∞)
which holds under, perhaps, eventual reordering of the elements of the orthonormal basis of H which is a complete orthonormal system for the separable Hilbert space H. If there is some integer α≥M such that the real sequence {|an|}n≥α converges to zero exponentially, then
(8)∥x-∑n=1Manen∥2=∑n=M+1∞|an|2=∑n=M+1α|an|2+∑n=α+1α|an|2≤C(α)+ρ1-ρ,
where |an|≤ρn≤ρ<1, for all n(∈N)≥α with C(α)=∑n=M+1α|an|2<+∞ being dependent on α such that C(α)=0. Hence, Property (vii).Note that Property (vi) of Lemma1 quantifies an approximation of an element of a finite-dimensional Hilbert space H via an orthonormal system in H of smaller dimension than that of such a space. Property (vii) relies on the approximation of an element in an infinite-dimensional separable Hilbert space by using a numerable orthonormal basis of H.Lemma 2.
LetT:H→H be a linear, closed, and compact self-adjoint operator in an infinite-dimensional separable Hilbert space H with a numerable orthonormal basis of generalized eigenvectors {en}n=1∞T:H→H. Then, the following properties hold:(i)
TNx=∑n=1∞λnN(T)〈x,en〉en,
for allx∈H for any N∈N, where λn(T)∈σ(T); the spectrum of the operator T is defined by λn(T)=〈Ten,en〉, for all n∈N and λnN(T)=〈Ten,en〉N∈σ(TN) with |〈Ten,en〉|N→0 as n→∞, for all N∈N.
IfPn is the orthogonal projection operator of H on the one-dimensional subspace Dn generated by the eigenvector en then
(9)limn→∞(Pn(TNx))={0}(∈Dn);∀n,N∈N,∀x∈H.
If PΩi is the orthogonal projection operator of H on the nΩi-dimensional eigensubspace Ωi, then
(10)limi→∞(PΩi(TNx))={0}(∈Ωi);∀N∈N,∀x∈H
with TNx=PΩi(TN)(x)⊕(I-PΩi(TN))(x) where PΩi(TN)(x)≡PΩi(TNx), for all n,N∈N, for all x∈H.(ii)
If, in addition,∥T∥≤α<1, then
(11)limN→∞|λn(T)|N=0,∑n=1∞|λn(T)|N≤αN1-αN<∞;∀n∈N,limN→∞(Pi(TNx))={0}(∈Di),limN→∞(PΩi(TNx))={0}(∈Ωi);∀i∈N.Proof.
Note that there is a numerable orthonormal basis forH since H is separable and infinite dimensional. Such a basis {en}n=1∞ can be chosen as the set of generalized eigenvectors of the linear self-adjoint T:H→H since it is closed and compact and then bounded
(12)x=∑n=1∞〈x,en〉en;∀x∈H.
Also, since the linear operatorT:H→H is closed and compact, the spectrum σ(T) of T:H→H is a proper nonempty (since T:H→H is infinite dimensional and bounded since it is compact) subset of C and numerable and it satisfies σ(T)=σp(T)∪{0}, with σc(T)∪σr(T)={0}, where σp(T), σc(T), and σr(T) are the punctual, continuous, and residual spectra of T:H→H, respectively. Note that {0}∈σ(T) is also an accumulation point of the spectrum σ(T) since H is infinite dimensional and T:H→H is compact. Also, since H is separable, the spectrum of T:H→H is numerable, and 〈ej,en〉=δjn; for all j,n∈N, one gets
(13)Ten=∑j=0∞〈Ten,ej〉en=∑j=0∞〈Ten,ej〉enδjn=〈Ten,en〉en=λn(T)en;∀n∈N,
where λn(T)=〈Ten,en〉 is an eigenvalue of T:H→H; that is, λn(T)∈σ(T), associated with the eigenvector en since
(14)λn(T)en=λn(T)〈en,en〉en=〈λn(T)en,en〉en=〈Ten,en〉=〈Ten,en〉en
so that
(15)Tx=∑n=1∞〈Tx,en〉en=∑n=1∞〈T(∑j=1∞〈x,ej〉ej),en〉enδjn=∑n=1∞〈T(〈x,en〉en),en〉en=∑n=1∞λn(T)〈x,en〉en;∀x∈H,
so that, except perhaps for reordering, |λn(T)|≥|λn+1(T)|, for all n∈N with {λn(T)}→0 since H is separable and σ(T) is numerable. Assume that for any positive integer N the following identity is true:
(16)TNx=∑n=1∞λnN(T)〈x,en〉en.
Then, since {en}n=1∞ is an orthonormal basis of generalized eigenvectors,
(17)TN+1x=T(TNx)=∑n=1∞λn(T)〈TNx,en〉en=∑n=1∞λn(T)〈∑j=1∞λjN(T)×〈x,ej〉ej,en∑j=1∞λjN〉en=∑j=1∞∑n=1∞λjN(T)λn(T)〈〈x,ej〉ej,en〉en=∑j=1∞∑n=1∞λjN(T)λn(T)×〈〈x,ej〉ej,en〉δjnen=∑n=1∞λnN+1(T)〈x,en〉en=∑n=1∞〈Ten,en〉N+1〈x,en〉en,
where δjn is the Kronecker delta. Then, λnN(T)=〈Ten,en〉N∈σ(TN). Furthermore, TN:H→H is compact as it follows by complete induction as follows. Assume that TN:H→H is compact, then it is bounded. Note also that TN:H→H is self-adjoint by construction and then normal. Thus, TN+1=T(TN):H→H is compact since it is a composite operator of a bounded operator TN:H→H with a compact operator T:H→H. Then, by complete induction, λnN(T)=〈Ten,en〉N→0  (∈σ(TN)) as n→∞, for any N∈N since TN:H→H is compact and H is infinite dimensional. Also,
(18)Pn(TNx)=〈Ten,en〉N-1Pn(x)=〈Ten,en〉Nen=λnN(T)en⟶0asn⟶∞;∀N∈N;∀x∈H,
where Pn is the projection operator of H on the one-dimensional subspace Dn generated by the eigenvector en so that Pnx=〈x,xn〉xn→0 as n→∞, for all x∈H. Thus, Property (i) has been proved. To prove Property (ii), take an orthonormal basis associated with the set of finite-dimensional eigenspaces of the respective eigenvalues. Note from Cauchy-Schwarz inequality that
(19)|λn(T)|N=|〈Ten+qn,en+qn〉|N≤∥T∥N∥en+qn∥N≤∥T∥N≤αN<1;∀n,N∈N
for some real constant α∈(0,1), where {qn}n∈N is a nondecreasing sequence of finite nonnegative integers defined by qi=∑j=1i-1pj being built such that each qn for n∈N accounts for the total of the dimensions pj of the eigenspaces Ωj associated with the set of eigenvalues {λ1(T),λ2(T),…,λn-1(T)}previous toλn(T) for n∈N after eventual reordering by decreasing moduli. Then, limN→∞|λn(T)|N=0, for all n∈N, and
(20)∥Pi(TNx)∥=|〈Tei+qi,ei+qi〉N|∥Pi(x)∥=∥〈Tei+qi,ei+qi〉N(∑j=0pi-1γi+qi(j)ei+qi(j))∥=∥λiN(T)ei+qi∥≤PipiαN,
where {ei+qi(j):j=0,1,…,pi-1} is now a set of pi linearly independent elements belonging to the orthonormal basis of H that generate the eigenspace Ωi associated with λi(T) with ei+qi(0)=ei+qi being an eigenvector and {γi+qi(j):j=0,1,…,pi-1} is a set of complex coefficients. Then, αNei→0 as  →∞, for all i∈N from (20), so that limN→∞(Pi(TNx))={0}(∈Di). If there are some multiple eigenvalues, with all being of finite multiplicity since the operator T:H→H is compact, the above expression may be reformulated with projections on the finite-dimensional eigenspaces associated to each of the eventually repeated eigenvalues leading to limN→∞(PΩi(TNx))={0}(∈Ωi), for all i∈N. Note that Ωi≡Dqi×Dqi…pi  ×  Dqi is the finite pi(≥1)-dimension of the eigenspace Ωi associated with λi(T), where pi is one-dimensional if λi∈σ(T) is single. Finally, it follows from (19) that
(21)∑n=1∞|λn(T)|N=∑n=1∞|〈Ten+qn,en+qn〉|N≤∑n=1∞αnN=αN1-αN<∞
and Property (ii) has been proved.Lemma2 becomes modified for compact operators on a finite-dimensional Hilbert space as follows.Lemma 3.
LetT:H→H be a linear closed and compact self-adjoint operator in a finite-dimensional Hilbert space H of finite dimension p with a finite orthonormal basis of eigenvectors {en}n=1p of T:H→H. Then, the following properties hold.(i)
TNx=∑n=1pλnN(T)〈x,en〉en
for anyN∈N, where λn(T)∈σ(T); the spectrum of the operator T is defined by λn(T)=〈Ten,en〉, for all n∈p- and λnN(T)=〈Ten,en〉N∈σ(TN), for all N∈N.(ii)
If, in addition,∥T∥N≤ηαN for some real constants α∈(0,1) and η≥1, then
(22)|λn(T)|N≤ηαN1-αN<∞,∀N∈N,|λn(T)|N⟶0asN⟶∞;∀n∈p-∑n=1p|λn(T)|N≤η(αN-αN(p+1))1-αN<∞;∀N∈N,∑n=1p|λn(T)|N⟶0asN⟶∞,∀p∈N.Outline of Proof. First note that the spectrum of T:H→H is nonempty since the operator is self-adjoint. Note also that, since the Hilbert space is finite-dimensional Hilbert space, any set of normalized linearly independent eigenvectors of a self-adjoint operator is an orthonormal basis of such a Hilbert space [1]. Property (i) is a direct counterpart of Property (i) of Lemma 2 except that {0} can be a value of the punctual spectrum of T:H→H but it is not an accumulation point of such a spectrum σ(T) since the Hilbert space is finite-dimensional. Therefore, the result 〈Ten,en〉→0 as n→∞ of Lemma 1 does not hold. Then, Property (i) follows directly from the above considerations. Property (ii) follows from the relations
(23)∑n=1p|λn(T)|N=∑n=1p|〈Ten+qn,en+qn〉|N≤∑n=1pηαnN=η(αN-αN(p+1))1-αN<∞.Remark 4.
It turns out that Lemma2 (ii) and Lemma 3 (ii) also hold if T:H→H is not self-adjoint since the corresponding mathematical proofs are obtained by using an orthonormal basis formed by all linearly independent vectors generating each of the subspaces. However, if the operator is not self-adjoint or if it is infinite dimensional while being self-adjoint, the set of (nongeneralized) eigenvectors is not always an orthogonal basis of the Hilbert space.In the following, we relate the properties of operators onH with their degenerate versions obtained via truncations of their expanded expansions.Theorem 5.
LetH be a separable Hilbert space and let T(p):H→H be a linear degenerated p-finite-dimensional approximating operator of the linear closed and compact self-adjoint operator T:H→H. Then, the following properties hold.(i)
Assume that∥T∥N≤ηαN, for all N∈N for some real constants α∈(0,1) and η≥1, where {en}n=1∞ is a numerable orthonormal basis of generalized eigenvectors of  T:H→H. Then,
(24)sup(∥TNx-TN(p)x∥:∥x∥≤1)≤ηαN(p+1)1-αN,∥TNx-TN(p)x∥⟶0asN⟶∞;∀x∈H.(ii)
Assume that there is a finiten0∈N such that ∑n=n0∞|λn(T)|≤M0<+∞ for some positive real constant M0=M0(n0). Thus, for any given positive real constant ε≤1, there are nonnegative finite integers p0=p0(ε,n0)>n0 and N0=N0(p0,ε) such that for any finite p(≥p0)-dimensional degenerated approximating operator T(p):H→H of T:H→H, the following inequality holds
(25)∥TN(p)x∥≤∥TNx∥+ε∥x∥≤(∥TN∥+ε)∥x∥;∀N>N0∀x∈H.Furthermore,
(26)limN→∞(∥TNx-TN(p)x∥)=0;∀x∈H
for any T(p):H→H linear degenerated p(≥p0)-finite-dimensional approximating operator of the linear closed and compact self-adjoint operator T:H→H and some finite p0∈N.(iii)
If{TNx}→z as N→∞ for some x,z∈H such that limN→∞(∥TNx-TN(p)x∥)=0, then {TN(p)x}→z as N→∞. Furthermore, such a z is a fixed point of both T:H→H and T(p):H→H.Proof.
The operatorT:H→H is represented as follows:
(27)Tx=∑n=1∞〈Tx,en〉en=∑n=1∞∑j=1∞〈〈Tx,ej〉ej,en〉en=∑n=1∞λn(T)〈x,en〉en.
The associated degenerated p-finite-dimensional operator is
(28)T(p)x=∑n=1p〈Tx,en〉en=∑n=1pλn(T)〈x,en〉en
so that
(29)TN(p)x=∑n=1p〈TNx,en〉en=TNx-∑n=p+1∞〈TNx,en〉en=∑n=1∞〈TNx,en〉en-∑n=p+1∞〈TNx,en〉en.
Thus, assume that TN(p)x=∑n=1pλnN(T)〈x,en〉en. Then,
(30)TN+1(p)x=∑n=1p〈TN+1x,en〉en=∑n=1p∑j=1∞〈〈TN+1x,ej〉ej,en〉en=∑n=1p〈〈TN+1x,en〉en,en〉en=∑n=1p〈〈λn(T)TNx,en〉en,en〉en=∑n=1p〈〈TNx,en〉en,en〉λn(T)en=∑n=1pλn(T)〈TNx,en〉en=∑n=1pλn(T)〈x,(TN)*en〉en=∑n=1pλn(T)〈x,(T*)Nen〉en=∑n=1pλnN+1(T)〈x,en〉en
so that the assumption TN(p)x=∑n=1pλnN(T)〈x,en〉en is true as it has been proved from (30) by complete induction. The following properties are also direct for any x∈H if ∥T∥N≤ηαN<1 for some real constants α∈(0,1) and η≥1; for all N≥N0 and some finite N0∈N, we have
(31)∥TNx∥=∥∑n=1pλnN(T)〈x,en〉en+∑n=p+1∞λnN(T)〈x,en〉en∥≤∥∑n=1pλnN(T)〈x,en〉en∥+ηαN(p+1)1-αN∥x∥=∥∑n=1pλnN(T)〈x,en〉en∥+ηαN(p+1)1-αN∥x∥=∥TN(p)x∥+ηαN(p+1)1-αN∥x∥∥TNx-TN(p)x∥=∥∑n=p+1∞〈TNx,en〉en∥≤ηαN(p+1)1-αN,∀x∈Hwith∥x∥≤1∥TNx-TN(p)x∥⟶0asN⟶∞;∀x∈H.
Property (i) has been proved. On the other hand, if ∑n=n0∞|λn(T)|≤M0<+∞ for some finite n0∈N and some M0∈R+, then for any given real ε(≤1) ∈R+, there is a positive finite integer p0=p0(ε)>n0 such that for any ν(∈R+)≤ε/M0 and any p(≥p0) ∈N, the following inequalities hold:
(32)∑n=p+1∞|λn(T)|≤∑n=p0+1∞|λn(T)|≤νM0≤ε≤∑n=n0+1∞|λn(T)|<∑n=n0∞|λn(T)|≤M0
since |λn(T)|≥|λn+1(T)|, for all n∈N, λn(T)→0 as n→∞, 0∈σ(T), and |λn(T)|≤ε, for all n(≥n0) ∈N. Note that since M0∈R+ exists such that ∑n=n0∞|λn(T)|≤M0<+∞ for some finite n0∈N, then, for any given ε(≤1) ∈R+, (32) holds for any p(≥p0)∈N and some p0=p0(ε)>n0. Then, one gets via complete induction for any N(>N0)∈N(33)∑n=p+1∞|λnN(T)|≤|λp+1(T)|(∑n=p+1∞|λnN-1(T)|)≤εN<1
and ∑n=p+1∞|λnN(T)|→0 as N→∞ if ε<1, for all p(≥p0) ∈N. Thus, one gets from Lemma 1 (iv)
(34)∥TNx∥=∥∑n=1pλnN(T)〈x,en〉en+∑n=p+1∞λnN(T)〈x,en〉en∥≤∥∑n=1pλnN(T)〈x,en〉en∥+∥∑n=p+1∞λnN(T)〈x,en〉en∥≤∥∑n=1pλnN(T)〈x,en〉en∥+(∑n=p+1∞|λnN(T)|)×∥∑n=p+1∞〈x,en〉en∥≤∥∑n=1pλnN(T)〈x,en〉en∥+(∑n=p+1∞|λnN(T)|)∥∑n=1∞〈x,en〉en∥≤∥∑n=1pλnN(T)〈x,en〉en∥+ε∥x∥,∀x∈H
for any p(≥p0) ∈N and for all N(>N0) ∈N. Furthermore, note from (32) that ν→0 and p0→∞ as ε→0 and the function ν=ν(ε) is nonincreasing. Also, a strictly monotone decreasing positive real sequence νn=ν(εn) can be built with {εn}→0 since there are infinite many values of the spectrum σ(T) such that the inequality |λn(T)|≥|λn+1(T)| is strict since, otherwise, the convergence of the sequence {|λn(T)|} to zero would be impossible. Then, from (34) and ∑n=p+1∞|λnN(T)|→0 as N→∞ if ε<1, for all p(≥p0) ∈N, there are subsequences of positive real and positive integers {εp0}→0 and {po(εp0)}→+∞, respectively, as N→∞ such that the following subsequent relation holds:
(35)∥TNx-∑n=1p≥p0λnN(T)〈x,en〉en∥=∥∑n=p+1∞λnN(T)〈x,en〉en∥≤εp0∥x∥,∀x∈H
for all N(>N0)∈N. Then,
(36)limsupN→∞(∥TNx-∑n=1p≥p0λnN(T)〈x,en〉en∥)≤0
and Property (ii) follows directly.
If{TNx}→z and limN→∞(∥TNx-TN(p)x∥)=0 as N→∞ for some x,z∈H, then ∃{z~N} which converges to zero such that
(37)0=limN→∞∥TNx-TN(p)x∥=∥z+z~N-TN(p)x∥≥|∥z-TN(p)x∥-∥z~N∥|≥|limsupN→∞(∥z-TN(p)x∥)-limN→∞∥z~N∥|=limsupN→∞(∥z-TN(p)x∥)
and then ∃limN→∞(∥z-TN(p)x∥)=0. Also, T:H→H is bounded, since it is compact, and it is then continuous since it is linear and bounded. Also, T(p):H→H is of finite-dimensional and closed image, then compact, and then bounded and continuous since it is linear. Thus, ∥z-TNx∥→0, ∥z-TN(p)x∥→0 as N→∞leads to
(38)0⟵∥TN+1x-z∥=∥T(TNx)-z∥⟶∥Tz-z∥asN⟶∞implyingz=Tz0⟵∥TN+1(p)x-z∥=∥T(p)(TN(p)x)-z∥⟶∥T(p)z-z∥asN⟶∞implyingz=T(p)z,
and Property (iii) has been proved.Note that Theorem5 (ii) cannot be generalized, in the general case, for the case of a finite dimensional approximating linear operator T(p):H→H of smaller dimension p<q to any linear degenerated operator T:H→H of (finite) dimension q. The reason is that the property that 0∈σ(T) does not any longer hold, in general if T:H→H is finite dimensional. On the other hand, a way of describing the operator T:H→H and its approximating finite-dimensional counterpart T(p):H→H is through the absolute error operator T-~p(≡T-T(p)): H→H. This is useful if either T:H→H is finite dimensional of dimension q>p where p is the dimension of  T(p):H→H or if T:H→H is nondegenerated. Another useful characterization is the use of the relative error operator T~(p):H→H satisfying the operator identity T(p)=T(I+T~(p)). Another alternative operator identity T=T(p)(I+T~1(p)) cannot be used properly if T:H→H is infinite dimensional since T(p):H→H is degenerated of finite dimension p. We discuss some properties of the operator identity T(p)=T(I+T~(p)) through the subsequent result.Lemma 6.
LetH be a separable Hilbert space and let T:H→H be a nonnull and nondegenerated (i.e., of infinite-dimensional image) linear closed and compact operator and let T(p):H→H be the linear degenerated p-finite-dimensional approximating operator of T:H→H. Then, there is an operator T~(p):H→H such that T(p) can be represented by T(p)=T(I+T~(p)), Dom(T~(p))⊆Dom(T), and Im(T~(p))⊆Dom(T) with the following properties.(i)
There exists an (in general, nonunique) operatorT~(p):H→H, restricted to T~(p):DomT~(p)|DomT→ImT~(p)⊆DomT for each approximating T(p):H→H of given dimension p.(ii)
The operatorTT~(p):H→H is nondegenerated, unique, and compact.(iii)
The minimum modulus ofT:H→H is μ(T~(p))=0 so that if it is invertible, its inverse is not bounded. If T:H→H is degenerated, that is, finite dimensional of dimension q>p, injective with closed image then its minimum modulus is positive and finite. If, furthermore, T:H→H is invertible then T~(p):H→H is a compact operator with bounded minimum modulus μ(T~(p)).Proof.
The existence of such an operatorT~(p):H→H is proved by construction. Let {en}n∈N be an orthonormal basis of generalized eigenvectors of T:H→H and {vn}n∈N an orthonormal basis of T~(p):H→H, respectively. Then, one gets for some sequences of complex coefficients {γnj}j∈N, for all n∈N,
(39)vn=∑j=1∞γnjejTx=∑n=1∞〈Tx,en〉en=∑n=1∞λn(T)〈x,en〉enT(p)x=∑n=1p〈Tx,en〉en=∑n=1pλn(T)〈x,en〉enT~(p)x=∑n=1∞〈T~(p)x,vn〉vn=∑n=1∞λn(T~(p))〈x,vn〉vn=∑n=1∞∑j=1∞λn(T~(p))〈x,γnjej〉vn=∑n=1∞∑j=1∞∑k=1∞λn(T~(p))〈x,γnjej〉γnkekδjk=∑n=1∞∑j=1∞λn(T~(p))〈x,ej〉γ-njγnjej=∑n=1∞∑j=1∞λn(T~(p))|γnj|2〈x,ej〉ej(40)TT~(p)x=T(∑k=1∞∑j=1∞λk(T~(p))|γkj|2〈x,ej〉ej)=∑n=1∞λn(T)〈∑k=1∞∑j=1∞λk(T~(p))|γkj|2×〈x,ej〉ej,en∑k=1∞∑j=1∞λk(T~(p))|γkj|2〉enδjn=∑n=1∞∑k=1∞∑j=1∞λn(T)λk(T~(p))|γkj|2〈x,ej〉ejδjn=∑n=1∞λn(T)(∑k=1∞λk(T~(p))|γkn|2)〈x,en〉en.
Then, TT~(p):H→H is a unique nondegenerated compact operator from its representation (40). It follows that the operator identity T(p)=T(I+T~(p)) holds on H if and only if T(p)x=T(Ι+T~(p))x; for all x∈H and, equivalently,  since T and T~(p) are linear,
(41)∑n=1∞λn(T)(1+∑k=1∞λk(T~(p))|γkn|2)〈x,en〉en=∑n=1pλn(T)〈x,en〉en.
Since the vectors in{en}n∈N form an orthonormal basis, (41), if the following constraints defining the operator T~(p):H→H, restricted as T~(p):DomT~(p)∣DomT→ImT~(p)⊆DomT, hold for a nonnull operator T:H→H(42)∑n=1p∑k=1∞λn(T)λk(T~(p))|γkn|2〈x,en〉en+∑n=p+1∞λn(T)(1+∑k=1∞λk(T~(p))|γkn|2)〈x,en〉en=0
so that (42) holds if and only if
(43)∑k=1∞λk(T~(p))|γkn|2=0forn∈p-;1+∑k=1∞λk(T~(p))|γkn|2=0forn>p
since the elements of {en} are linearly independent. Then (43) holds under infinitely many combinations of constraints on the spectrum of T~(p):H→H. In particular, (43) holds if
(44)λn(T~(p))=0,∀n(∈N)≠p+1;λp+1(T~(p))=-1|γp+1,n|2;|γp+1,n|=0forn∈p-|γp+1,n|=γ≠0forn(∈N)>p+1.
Equations (43) are also satisfied with γkn=0, for all n∈p-, for all k∈N, and 1+∑k=1∞λk(T~(p))|γkn|2=0 for n>p which holds, for instance, if |γkn|2=|γn|2=-1/(∑k=1∞λk(T~(p))) for all n>p. Thus, T~:H→H is then non-unique, in general. Properties (i)-(ii) have been proved.
Now, letμ(Γ)={inf∥Γx∥:x∈H,∥x∥=1} be the minimum modulus of the linear operator Γ:H→H. If ∥x∥=1, then if T:H→H is injective with closed image (this implies that such an image is finite dimensional), then μ(T)>0 and since T,T-:H→H are both bounded since they are compact, one gets
(45)μ(T~(p))≤μ(TT~(p))μ-1(T)≤max∥x(∈H)∥=1∥Tx-T(p)x∥μ-1(T)≤∥T-T(p)∥μ-1(T)≤(∥T∥+∥T(p)∥)μ-1(T)<∞.
If T:H→H is infinite dimensional, then μ(T)=0 and it cannot then have bounded inverse. If T:H→H is degenerated of dimension q=p, then T~(p) is the null operator with μ(T~)=0. If T:H→H is degenerated of dimension q>p and invertible, then μ-1(T)=μ-1(T*)=∥T-1∥<∞ and ∥T~(p)∥≤∥T-1∥∥T-T-∥<∞ so that T~(p):H→H is bounded and compact since it is a composite operator of a compact operator (T-T(p)) on H and a bounded operator T-1 on H. Property (iii) has been proved.Example 7.
Assume thatT,T(p):H→H are two degenerated finite-dimensional operators on a separable Hilbert space H of, respectively, dimensions two and one defined by Tx=λ1(T)〈x,e1〉e1+λ2(T)〈x,e2〉e2; for all x∈H and T(p)x=λ1(T)〈x,e1〉e1; for all x∈H. Thus, the constraints (42) hold for an incremental operator T~(p):H→H of spectrum defined by λ1(T~(p))=0, λ2(T~(p))=-1/|γ22|2 with γ22≠0 and γ21=0. Then,
(46)T~(p)x=-1|γ22|2〈x,e2〉e2;TT~(p)x=-λ2(T)λ2(T~(p))|γ22|2〈x,e2〉e2=-λ2(T)〈x,e2〉e2.Remark 8.
IfT:H→H is infinite dimensional and invertible, then T~(p):H→H is not compact, since T-1:H→H is unbounded, since μ(T)=0⇔μ-1(T)=∥T-1∥=∞.


## 3. Examples

Hilbert spaces for the formulation of equilibrium points, stability, controllability [16, 18, 19], boundedness, and square integrability (or summability in the discrete formalism) of the solution in the framework of square-integrable (or square-summable) control and output functions are of relevant importance in signal processing and control theory and in general formulations of dynamic systems, in general. See, for instance, [1, 2, 7, 9, 16, 17, 19, 20] and the references therein. Two examples with the use of the above formalism to dynamic systems and control issues are now discussed in detail.Example 1.
Consider the forced linear time-invariant differential system of real coefficients andnth as
(47)∑i=0sαndiy(t)dti=βu(t)
under a piecewise continuous square-integrable forcing function u:R0+→R; that is, u∈L2(0,∞), with αn≠0. The unique solution for any given initial conditions (dyi(0))/dti for i=0,1,…,s-1 is
(48)y(t)=cT(eAtx(0)+βαn∫0teA(t-τ)bu(τ)dτ),
where the superscript Tstands for transposition, c,b∈Rs are Euclidean vectors of, respectively, first and last components being unity and the remaining ones being zero x(t)=(y(t),(dy(t))/dt,…,ds-1y(t)/dts-1)T, and
(49)A=[010⋯000010⋯0⋯⋯⋯⋯01-αn-1αn-αn-2αn⋯-α0αn].
The matrix functioneAt is a C0-semigroup generated by the infinitesimal generators A, respectively [17, 19]. Using a sampling period of length θ, we can write from (48) for time instants being integer multiples of the sampling period
(50)xn+1≔x((n+1)θ)=T(u,θ,n)xn=Th(θ)xn+Tf(θ)u(n,θ)=eAθ(xn+βg∫0θe-Aτbu×(nθ+τ)dτβg∫0θe-Aτ),∀n∈N0=N∪{0},yn+1≔cTx((n+1)θ)=cTeAθ×(xn+βg∫0θe-Aτbu(nθ+τ)dτ),∀n∈N0,
where xn=x(nθ) and g=αn provided that the input is un=un(θ)=un(τ), for all τ∈[nθ,(n+1)θ). The matrix function eAθ can be expanded as follows:
(51)eAθ=∑k=0μ-1αk(θ)Ak=∑k=0μ-1(∑j=0νk-1∑ℓ=0ϑγjkθjeλℓθ)Ak,
where σ(A)={λk:k=0,1,…,ϑ-1} is the spectrum of A, that is, set of ϑ distinct eigenvalues of A with respective multiplicities νk for k=0,1,…,μ-1 in the minimal polynomial of A where μ=∑k=1ϑνk is the degree of the minimal polynomial of A, and then 1≤μ≤s and γjk are complex constants. The above αk(t); k=0,1,…,μ-1 are everywhere continuous and linearly independent time-differentiable functions on R. Then, the unique solution (or output) of (47) for zero initial conditions is
(52)y(t)=(Λcu)(t)=∫0∞h(t,τ)u(τ)dτ=∫0th(t-τ)u(τ)dτ
with y∈L2(0,∞) provided that h∈L2(0,∞), guaranteed from (51) if and only if Re(λ)<0; for all λ∈σ(A) and h(t,τ)=h(t-τ) is a convolution operator and Λc is a convolution integral operator since the differential system is time-invariant where h(t,τ)=0, for all τ(>t)∈R0+. Thus, such an operator is normal, since it is time invariant [1], and then self-adjoint. Now, define the sequence of samples {yn:=y(nθ)}n∈N for a sampling period θ as
(53)yn≔Λu^n=(Λcu)(nθ)=∫0nθh(nθ,τ)u(τ)dτ;∀n∈N
with the operator Λ being defined from Λc on the space of square-summable sequences ℓ2(0,∞), where u^n:=(u0,u1,…,un-1), for all n∈N. Assume that the forcing input u(t)=un=u(nθ) is piecewise constant, for all n∈N, for all t∈[nθ,(n+1)θ). Note that if h0=0, then hL(s), the unilateral Laplace transform of h(t), is strictly proper; that is, it has more poles than zeros. In the case that h0=h(0)≠0,hL(s) is proper by not strictly proper; that is, it has the same number of poles and zeros. It turns out that we can define an operator sequence T^n:ℓ2[0,∞)→ℓ2[0,n+1]: for all n∈N, with the second one being a natural projection Pn+1 on ℓ2[0,n+1] of an operator T^ on ℓ2[0,∞] so that, by using T^n:ℓ2[0,∞)→ℓ2[0,n+1]; for all n∈N, one gets:
(54)y^n+1=T^ny^n=T^y^;∀n∈N
with y^n=(y0,y1,…,yn,0,0…)T; for all n∈N, y^=(y0,y1,…,yn,yn+1,yn+2,…)T, y^0=y0=T0y0, with T0 being the identity operator. One has from (51) that
(55)hn=h(nθ)=βcTg(∑k=0μ-1∑j=0νk-1∑ℓ=0ϑγjk(nθ)jeλℓnθAk)b;∀n∈N0
and hn→0 as n→∞ if Re(λi)<0; ℓ=0,1,…,ϑ-1. Some particular cases are discussed below under the assumption {un}⊂ℓ2[0,∞) and Re(λi)<0; ℓ=0,1,…,ϑ-1 implying {hn}⊂ℓ2[0,∞), {|hn|}⊂ℓ[0,∞), so that ∑i=0n|hi|=H-<+∞ and ∑i=0nhi=H<+∞, since {hn} is bounded.Proposition 2 (constant piecewise constant open-loop control).
Assume thatRe(λ)<0, for all λ∈σ(A), and consider a constant open-loop control un=u0, for all n∈N. The following properties hold.(i)
The sequence{yn}n∈N0+ satisfies yn+1=Tyn=Tn+1y0, subject to y0=u0h0, for all n∈N0, where the operator T:N0×R→R is defined as the sequence of scalar gains {∑i=0n+1hn+1-i/∑i=0nhn-i}, for all n∈N in the Banach space (R,|·|) which is the Euclidean Hilbert space for the product of real numbers being an inner product. Furthermore, {yn}n∈N0→y*.(ii)
Assume thatp(∈N)≥p0 for some given p0∈N, and |u0|<min(1/H-,1/(∑i=0p0-1|hn-i|)). Then, |yn|N→0 and |y-n(p)|N→0 as N→∞, for all n∈N, for all p(≥p0) ∈N for some finite p0∈N.(iii)
There isp0=p0(ε,u0)∈N for each given ε∈R+ and u0∈R such that |yn-y-n(p)|≤ε, for all n(∈N0)≥p≥p0. Also, for each given u0∈R satisfying ∃limN→∞(|u0|max0≤i≤p-1(|hn-i|))N=0, for all n(∈N)≥p-1, it follows that
(56)limN→∞|yn-y-n(p)|N=0,limN→∞(ynN-y-nN(p))=0,∀n(∈N)≥p-1.Proof.
Property (i) follows fromyn=u0(∑i=0nhn-i), or equivalently, yn+1=(∑i=0n+1hn+1-i)/(∑i=0nhn-i)yn, for all n∈N0 subject to an initial condition y0=u0h0. Since {hn} is bounded, {hn}→0 as n→∞, and ∑i=0nhi=H<+∞, then yn=u0(∑i=0nhn-i)→y*=u0H=u0(∑i=0∞hn-i)<+∞. Thus, the sequence {yn}n∈N0+ is generated as yn+1=Tyn=Tn+1y0, subject to y0=u0h0, for all n∈N0, where the operator T:N0×R→R is defined in the Banach space (R,|·|) as the sequence of scalar gains {(∑i=0n+1hn+1-i)/(∑i=0nhn-i)}, for all n∈N which is the Euclidean Hilbert space for the product of real numbers being an inner product. Furthermore, {yn}n∈N0→y*. Property (i) has been proved. On the other hand, since |u0|<min(1/H-,1/(∑i=0p0-1|hn-i|)), it follows that |y*|≤|u0H-|<1 and then
(57)|yn|=|u0|(|∑i=0nhn-i|)<|∑i=0nhn-i|∑i=0∞|hi|≤1∀n∈N,|yn|⟶|y*|<1asn⟶∞,|yn|N⟶0asN⟶∞,∀n∈N,(58)|y-n(p)|=|u0||∑i=0p-1hn-i|<∑i=0p-1|hn-i|∑i=0p0-1|hn-i|≤∑i=0p-1|hn-i|∑i=0p-1|hn-i|=1,∀n∈N,|y-n(p)|N⟶0asN⟶∞,∀n∈N.
Property (ii) has been proved. Now, note that for any givenε∈R+ and u0∈R, there is p0=p0(ε,u0)∈N such that for any p(≥p0) ∈N(59)|yn-y-n(p)|=|u0||∑i=p+1nhn-i|≤|u0|(∑i=p0+1n|hn-i|)≤|u0|(∑i=p0+1∞|hn-i|)≤ε
for p0=p0(ε,u0)∈N satisfying (∑i=p0+1∞|hn-i|)≤ε|u0|-1 if u0≠0 and such a p0 exists since {|hn|}⊂ℓ[0,∞). Note that if u0=0, then |yn-y-n(p)|=0 so that |yn-y-n(p)|≤ε for any p0=p0(ε)∈N. The first part of Property (iii) has been proved. Note that ∃limN→∞|ynN-y-nN(p)|=0. Then, the second part of Property (iii) follows since
(60)limsupN→∞|ynN-y-nN(p)|≤H|u0|limsupN→∞(|u0|N-1max0≤i≤p|hn-i|N-1)=0.
Property (iii) follows from Theorem 5 with the operator T^n:ℓ2[0,n]→ℓ2[0,n+1], for all n∈N of (58) and its degenerated finite truncation Tn^-(p):ℓ2[0,p], for all n∈N in the subsequent way
(61)|yn-y-n(p)|=∥y^n-y^n(p)∥=∥T^n-1y^n-1-T^n-1(p)y^n-1∥=∑i=1∞〈|Tny0-T-n(p)y0|,|u0|1/2givi〉×|u0|1/2givi,|ynN-y-nN(p)|=∥T^n-1y^n-1-T^n-1(p)y^n-1∥N=∑i=1∞〈|u0|1/2|Tny0-T-n(p)y0|,|Tny0-T-n(p)y0|N-1×|u0|1/2givi〉|u0|1/2givi,
where T:R→R maps each element of the sequence {yn}n∈N0, which is strictly ordered according to the time occurrence, to its next consecutive one,
(62)gi={hiifhi≥0i|hi|ifhi<0
(then gi2≠-hiin the second part of (62)), for all i∈N0, and {vi}i∈N is a basis of orthogonal vectors vi=|u0hi|-1/2ei if u0hi≠0 and vi=0, where ei is the ith unit vector in Rn with its ith component being one, such that the set {|u0|1/2|hi|vi}i∈N is an orthonormal basis so that 〈|u0|1/2|hi|vi,|u0|1/2|hj|vj〉=δij as
(63)y-n(p)=∑i=n-p+1nhn-iui=11-h0snn(∑i=n-p+1n-1∑j=0ihn-isijyj),∀n∈N.Example 2.
Consider again (47) with Re(λ)<0, for all λ∈σ(A). If one measures some more state variables than just the solution, then an extended solution (48) of the form
(64)z(t)=(y(t),x0T(t))T=C(eAtx(0)+∫0teA(t-τ)Bu(τ)dτ)
is built with z:R0+→Rs0which is the output; 1≤s0≤s, z(t)=Cx(t) and x0(t) is formed by all or some of the components of x(t) except y(t), C∈Rs0×s, and B∈Rs×sm where sm≥1 is the dimension of the piecewise-continuous input u:R0+→Rsm which is in Lsm2[0,∞). If x0(t) is not used to (64), then z(t)=y(t) and s0=1. If z(t)=x(t), then s0=s. Equation (64) can be expressed as
(65)z(t)=(Λhx0)(t)+(Λfu)(t)=Th(t)x0+∫0∞Tf(t,τ)u(τ)dτ=Th(t)x0+(T^fue)(t),∀t∈R0+
with x0=x(0), Th(t)=CeAt and the operators Tf:R0+2×Rsm→Rs0 and T^f:Lsm2(-∞,∞)→Ls02(-∞,∞) are defined as
(66)Tf(t,τ)=Tf(t-τ)=CeA(t-τ)B;∀t∈R0+(67)(T^fue)(t)=∫-∞∞CeA(t-τ)Bue(τ)wt(τ)dτ=∫0∞Tf(t-τ)u(τ)1(t-τ)dτ=∫-∞∞Tf(t-τ)ue(τ)1(t-τ)dτ=∫-∞∞Tf(t-τ)wt(τ)ue(τ)dτ;∀t∈R0+
so that Tf(t,τ)=0 for τ>t is a convolution operator, where ue:R→Lsm2(-∞,∞)∩PC(R,Rsm) is piecewise continuous on R and square integrable defined as ue(t)=u(t) for t∈R0+ and u-(t)=0; otherwise, wt is a truncated multiplicative (or truncated gate) from (-∞,t]  ∩  R to (0,1) for each defined as wt(τ)=1 for 0≤τ≤t and wt(τ)=0, otherwise, for all t∈R. Note that wt(τ)=1(t-τ)1(τ), for all (t,τ)∈R2. The multiplicative (or gate) operator w from R to (0,1) is defined as w(t)=1for t≥0 and w(t)=1, otherwise; for all t∈R. Now,
(68)(T^f(p)ue)(t)=∑i=1p〈(T^fu)(t),θi(t)〉φi(t)=∫-∞∞Tf(p,t-τ)ue(τ)1(t-τ)dτ=∫-∞∞Tf(p,t-τ)wt(τ)ue(τ)dτ,
where 〈·,·〉 is the inner product on Ls02(-∞,∞), Tf(p,t-τ) is the kernel of (T^f(p))(t); {θi} and {φi}; i∈p- are two reciprocal orthogonal bases of the pth dimensional subspace Mp of Ls02(-∞,∞) and T^f(p) maps ue∈Lsm2(-∞,∞) in the orthogonal projection of (T^fue)(t) on Mp, for all t∈R0+. Note that T^f(p) is a self-adjoint, since it is time invariant (and convolution), compact operator since its image is finite dimensional. On the other hand, note that
(69)∬-∞∞∥Tf(t-τ)wt(τ)∥2dτdt=∬-∞∞∥Tf(t-τ)∥2|wt(τ)|2dτdt=∥w∥∥tf∥<+∞
so that T^f:Lsm2(-∞,∞)→Ls02(-∞,∞) has a square-integrable kernel so that it is a Hilbert-Schmidt operator, then compact, and also self-adjoint since it is time invariant. Note that ∥ue∥=∥u∥. Thus,
(70)(T^fue)(t)=∫-∞∞CeA(t-τ)Bue(τ)wt(τ)dτ=∑i=1∞〈(T^fue)(t),θi(t)〉φi(t),
where {θi} and {φi}; i∈N are two orthogonal complete systems of the infinite-dimensional separable Hilbert space Ls02(-∞,∞) and one has from (67) to (69) that
(71)∬-∞∞∥(Tf(t-τ)-Tf(p,t-τ))wt(τ)∥2dτdt⟶0asp⟶∞(72)Tf(t-τ)=∑i=1∞ψi(t)θ-i(τ),Tf(p,t-τ)=∑i=1pψi(t)θ-i(τ),ψi(t)=∫-∞∞Tf(t-τ)φi(τ)dτ,∀t∈R0+∥((T^f-T^f(p))ue)(t)∥=∥∑i=p+1∞〈(T^fue)(t),θi(t)〉φi(t)∥,∀t∈R0+(73)sup∥u∥=1∥(T^f-T^f(p))ue∥=sup∥u∥=1∥∑i=p+1∞〈(T^fue),θi〉φi∥=|λp+1(T^f)|
with ue∈Lsn2(-∞,∞), λp+1(T^f)∈σ(T^f), being nonzero for any finite p, ψi:R→Ls02(-∞,∞); foralli∈N is a linearly independent set, since the kernel Tf(t-τ) of T^f is bounded and ψi:R→Ls02(-∞,∞), for all i∈N, and where the norm is associated with the inner product on Ls02(-∞,∞). Equation (70) describes the truncated error norm on (-∞,t] of (T^f-T^f(p))ue in (71), for all t∈R0+ via the formula (69) while (71) refers to the whole real interval (-∞,∞). From (73), there is p0=p0(ε) such that ∥T^f-T^f(p)∥≤ε for any p≥p0 and any prefixed ε∈R+. Since (T^fue)(t)=z(t)-Th(t)x0, (T^f(p)ue)(t)=zp(t)-Th(t)x0, for all t∈R0+, and |λn(T^f)|→0 as n→∞ since T^f:Lsm2(-∞,∞)→Ls02(-∞,∞) is compact and then
(74)∥(T^f-T^f(p))ue∥=∥∑i=p+1∞〈(T^fue),θi〉φi∥=|λp+1(T^f)|∥u∥≤ε∥u∥,∀p≥p0,∀t∈R0+,limsupt→∞∥(T^f-T^f(p))ue(t)∥=limsupt→∞∥z(t)-zp(t)∥≤|λp+1(T^f)|∥u∥,limp→∞(limsupt→∞∥(T^f-T^f(p))ue(t)∥)=limp→∞(limsupt→∞∥z(t)-zp(t)∥)≤limp→∞|λp+1(T^f)|=0,
concluding the following: (a) the true and approximate forced and complete solutions might be made as close as suited, in terms of difference of norms, by using a finite-range operator approximant of sufficiently large range dimension; (b) if the true asymptotic solution is a fixed point z*=∫0∞CeA(t-τ)Bu(τ)dτ, then
(75)limsupt→∞∥zp(t)∥2≤∥z*∥2+ε(p)∥u∥,liminft→∞∥zp(t)∥2≥|∥z*∥2-ε(p)∥u∥|,limp→∞limt→∞(∥zp(t)-z*∥2)=0,
so that zp(t)→z* as p(∈N),t(∈R)→∞, where ∥·∥2 denotes the spectral norm for vector and matrices. Now, assume that the dynamics is perturbed with a parametrical disturbance A~ in the matrix A, which is nonsingular since Reλ<0, for all λ∈σ(A) to give A′=A+A~=A(I+A-1A~), with I being the nth identity matrix. Thus, A′ is also a stability matrix if 1>∥A~∥∥A-1∥ for any matrix norm since from Banach perturbation lemma ∥A′-1∥≤∥A-1∥/(1-∥A-1∥∥A~∥) [7, 19, 21, 22], since A′-1=(I+A-1A~)-1A-1, exists and its maximum modulus eigenvalues do not cross the imaginary complex axis from the continuity of the eigenvalues with respect to the matrix entries. Thus, the perturbed dynamic system has the following solution:(76a)x(t)=eA′tx(0)+∫0teA(t-τ)(Bu(τ)+A~x(τ))dτ=eA′tx(0)+∫0teA′(t-τ)Bu(τ)dτ;∀t∈R0+,(76b)z(t)=(y(t),x0T(t))T=C(eA′tx(0)+∫0teA(t-τ)(Bu(τ)+A~x(τ))dτ)=C(eA′tx(0)+∫0teA′(t-τ)Bu(τ)dτ),∀t∈R0+.
If the nominal (i.e., unperturbed) solution is a fixed pointz* and, since ∥eA′t∥2→0 as t→∞ since A′ is a stability matrix, then applying Holder’s inequality to (76a) and (76b), it follows that x∈L∞ with supt∈R0+∥x(t)∥2≤M<+∞, and then(77a)limsupt→∞∥x(t)∥2≤∥z*∥2+KAρA(∥u∥),∀t∈R0+,(77b)limsupt→∞∥z(t)∥2≤∥z*∥2+δ∥C∥∥A∥2KAρAsupt∈R0+∥x(t)∥2,∀t∈R0+,(77c)limsupt→∞∥zp(t)∥2≤∥z*∥2+ε(p)∥u∥+δ∥C∥∥A∥2MKAρA,(77d)limp→∞(limsupt→∞∥zp(t)∥2)≤∥z*∥2+δ∥C∥∥A∥2MKAρA
for anyδ∈R+ satisfying ∥A~∥2≤δ<1/∥A-1∥2=λmin(ATA), and KA≥1 and ρA>0 are real constants such that ∥eAt∥2≤KAe-ρAt, for all t∈R0+. In particular, (-ρA) is the stability abscissa of the dominant eigenvalue of A if it is either simple or it has an associate diagonal Jordan block, or a number arbitrarily close to it but larger.



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*Source: 890657-2013-10-31.xml*