# On Norm-Attainable Operators in Banach Spaces

Authors: N. B. Okelo
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Norm-attainable operators have been studied over a period of time with nice results obtained particularly in Hilbert spaces. In this work, we consider the Banach space setting by characterizing nonpower operators onH and elementary operators. Lastly, we characterize a new notion of norm-attainability for power operators in general Banach spaces.


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## Body

## 1. Introduction

Studies on Hilbert space operators have elicited a lot of interest from mathematicians for decades. Characterizations of properties of Hilbert space operators have been done by many mathematicians over a long period of time with interesting results being obtained. Such properties include norms, Green function, numerical ranges, positivity, spectrum, and invertibility among others (see [1–5], [6] and the references therein). Norm-attainability is also a property which has been give keen attention. This property still remains very important as it has a lot of open questions which are unanswered particularly when a super class of Hilbert space operators called supraposinormal operators [3] is considered. In [7], the authors characterized the norm property for elementary operators and gave conditions under which a general elementary operator is norm attainable. In-depth characterization of norm-attainable operators has also been done in detail with considerations given to other properties like orthogonality (see [1–5] and the references therein). Regarding derivations, authors in [5] showed that if VΓ and WΓ are Γ-Banach algebras and δ an α-inner derivation, then δ is norm attainable if and only if the adjoint, δ∗, of δ is norm attainable. Moreover, as a consequence, they proved that if δN1 and δN2 are norm attainable then δN is norm attainable if either δN1 and δN2 or both are zero derivations and δN1 and δN2 are α-inner derivation and α′-inner derivation, respectively. On elementary operators, necessary and sufficient conditions for norm-attainability for Hilbert space operators were given in [2] where it was proved that if S∈BH,β∈W0S, and α>0, then there exists an operator Z∈BH such that ∥S∥=∥Z∥, with ∥S−Z∥<α. Furthermore, there exists a unit vector η∈H such that ∥Zη∥=∥Z∥ with Zη,η=β, where W0S denotes the maximal numerical range of the operator S. Moreover, norm-attainability conditions for elementary operators and generalized derivations have been given. For orthogonality of elementary operators in norm-attainable classes, a detailed exposition has been given in [4, 8]. A superclass of Hilbert space operators has also been considered in norm-attainable classes. In [3], a good characterization has been done on α-supraposinormality of operators in dense norm-attainable classes. In this paper, we continue in the spirit of characterization of operators in normed spaces. We characterize norm-attainability for functionals in Banach spaces. Moreover, we give a new notion of norm-attainability for power operators and also for elementary operators which generalizes the results of [2].


## 2. Preliminaries

In this section, we outline preliminary concepts which are useful in the sequel.Definition 1 [9].
LetV be a linear vector space. A nonnegative real valued function ∥.∥:V⟶ℝ is called a norm on V if it satisfies the following condtions:
(i)
∥x∥≥0 and ∥x∥=0, if and only if x=0, for all x∈V(ii)
∥αx∥=αx, for all x∈V and α∈K(iii)
∥x+y∥≤∥x∥+∥y∥, for all x,y∈VThe ordered pairV,. is called a normed linear space or simply a normed space.Remark 2.
Examples of normed spaces are Banach space, Hilbert space, Hardy space, Orlicz space, Sobolev-Folland-Stein space, andBH among others.Definition 3 [9].
An operatorA∈BH is called a scalar operator of order m if it possesses a spectral distribution of order m, i.e., if there is a continuous unital morphism ϕ:C0mℂ⟶BH such that ϕz=A, where z stands for the identity function on C and C0mℂ for the space of compactly supported functions on ℂ continuously differentiable of order m, 0≤m≤∞. An operator A0∈BH is called subscalar if it is similar to the restriction of a scalar operator to an invariant subspace.Definition 4 [1].
An operatorA∈BH is said to be normal if AA∗=A∗A and p-normal if ApA∗=A∗Ap; self-adjoint if A=A∗; positive if A=A∗ and Ax,x≥0, for all x∈H; and projection if A2=A=A∗.Definition 5 ([2], Definition 1.1).
An operatorA∈BH is said to be norm attainable if there exists a unit vector x0∈H such that ∥Ax0∥=∥A∥. The set of all norm-attainable operators on a Hilbert space H is denoted by NAH.Definition 6 ([2], Definition 1.2).
For an operatorA∈BH, we define a numerical range by WA=Ax,x:x∈H,∥x∥=1 and the maximal numerical range by W0A=β∈ℂ:Axn,xn⟶β,where∥xn∥=1,∥Axn∥⟶∥A∥.Definition 7 [10].
LetΩ be a Banach lattice then Ω is an abstract M space, i.e., Ω∈AM if x∧y=0 implies ∥x+y∥=max∥x∥,∥y∥. Also Ω is abstract L space, i.e., Ω∈AL if x∨y=0 implies ∥x+y∥=∥x∥+∥y∥.Definition 8 [9].
LetΩ be a Banach lattice. Then, Ω∈AM implies Ω∗∈AL and Ω∈AL implies Ω∗∈AM. Ω∈AM if and only if for any x,y∈Ω,x,y≥0 implies ∥x∨y∥=max∥x∥,∥y∥. Ω∈AL if and only if for any x,y∈Ω,x,y≥0 implies ∥x+y∥=∥x∥+∥y∥.Definition 9 [10].
LetΩ be a Banach lattice. Then, Ω is said to be σ complete, if for every order bounded sequence xn∈Ω,∨n≥1xn exists in Ω. Also, Ω is said to be bounded σ complete, provided that the any norm bounded and order monotone sequence in Ω is order convergent.Definition 10 [10].
LetΩ be a Banach space. An element x∈DΩ is called an extreme point of BΩ if x=λy+1−λz,y,z∈BΩ, and λ∈0,1, imply y=z. In this case, we write x∈extBΩ.Definition 11 [1].
LetA be a Banach algebra and consider T:A⟶A. The operator T is called an elementary operator if it has the representation TX=∑i=1nAiXBi,∀X∈A, where Ai,Bi are fixed in A or MA, where MA is the multiplier algebra of A.Example 12.
LetA=BH. For A,B∈BH, we define the particular elementary operators:
(i)
The left multiplication operatorLA:BH⟶BH by LAX=AX, for all X∈BH(ii)
The right multiplication operatorRB:BH⟶BH by RBX=XB, for all X∈BH(iii)
The generalized derivation (implemented byA,B) by δA,B=LA−RB, for all X∈BH(iv)
The basic elementary operator (implemented byA,B) by MA,BX=AXB, for all X∈BH(v)
The Jordan elementary operator (implemented byA,B) by UA,BX=AXB+BXA, for all X∈BH


## 3. Norm-Attainability for Functionals

In this section, we characterize norm-attainability of functionals in Banach spaces. We regardH∗ the dual space of a Hilbert space H to be nonzero throughout this section unless otherwise stated. Let φ∈H∗. Then, φ is said to be norm attainable at φ/∥φ∥ if there exists T∈BH such that φ,T=∥φ∥.∥T∥>0.φ∗/∥φ∗∥ is called a support for φ. The following proposition shows that any functional is norm attainable in nonzero dual spaces.Proposition 13.
LetBW be the set of all bounded linear maps on an Orlicz space W then every φ∈H+∗ is norm attainable on BW.Proof.
LetW be a bounded linear Orlicz space and bn∈W such that bn is monotone decreasing to 0. Given Jn∈BW, we have φJn>∥φ∥−bn since πJn≤1<∞ and φW0=0. Suppose that πJn≤2−n and let Jt=sup∣Jnt∣. Then, πJ≤∑n=1∞πJn≤1, that is, J∈BW and φJ≥supn∣Jn∣=∥φ∥, because φ∈H+∗.Theorem 14.
LetBW be the set of all bounded linear maps on an Orlicz space W then φ∈H∗ is norm attainable on BW if and only there exists L in a subspace C of BW such that φ+=φL and φ−=φGL.Proof.
It is known that we haveP,Q∈BW such that φLPφ∗∥φ+∥ and φG\LQ=∥φ−∥ by Proposition 13. Assume that πPL≤1/2 and πQG\L≤1/2. Then, πPL−QG\L≤πP+πQ≤1. Indeed, PL−QG\L∈LW and φPL−yG\L=∥φ+∥+∥φ−∥=∥φ∥. Suppose that P∈BW satisfies φP=∥φ∥. Let L=ł∈G:Pl≥0 then we show that φL,−φG\L∈H+∗. Now, if φL∈H+∗, then there exists Q∈W+. If φL∉H′, then there exists Q∈W+ such that φLQ<0. Now φ being singular, assume πQ≤1/2 and πP≤1/2. Then, J=PG\L−QL∈BW, and so, ∥φ−∥≥φ−−J=−φ+J+φJ≥φJ=φPG\L−φLQ>φPGL. This is contrary to our earlier assumption. Lastly, ∥φ∥=∥φ+∥+∥φ−∥>φ∣PL+φPG\L=φ∣P=∥φ∥.φG\L∈H1∗ follows analogously.Corollary 15.
Letφ∈H∗ be norm attainable at P∈BW, then φPA∥φA∥ for all A∈C.Proof.
∥φ∥=∥φA∥+∥φGA∥≥φAP+φG\AP=φP=∥φ∥ is enough.Theorem 16.
Letφ∈H∗ be singular then the set of all such φ is dense in H∗.Proof.
Letφ∈H∗ be singular and ε>0 be given. Then, we have L∈C such that ∥φ+G\L∥<ε and ∥φ−L∥≤ε. Suppose that ψ=φ+L−φ−G\L. Then, by Theorem 14, ψ is norm attainable. Also, ∥φ−ψ∥≤∥φ+−ψl∥+∥φ−ψG\l∥=∥φ+G\l∥+∥φ−l∥<2ε.Theorem 17.
LetBW be the set of all bounded linear maps on an Orlicz space W. Then, ϕ=χ+φ0≠χ∈H0∗,φ∈H∗ is norm attainable at P∈BW if and only if πP=1, φP=∥φ∥, and ∫GkχtPtdt=πx+ωkχ, where k∈KNχ=k:k−11+ωkχ=∥χ∥N0.Proof.
From the statement of the theorem, we have∥ϕ∥0=fP=k−1kχ,χ+φP≤k−1πP+ωkχ+φP≤k−11+ωkχ+∥φ∥=∥χ∥N0+∥ϕ∥=∥ϕ∥0. The converse follows from the fact that ϕ is singular from Theorem 16 and an assertion from Proposition 13.At this point, we consider norm-attainable functionals in Banach lattices. We denote an abstractL space and abstract M space by AL and AM, respectively. For details on AL and AM, see [10]. We state the following lemma.Lemma 18.
LetΩ∈AL and φ∈BΩ∗. Then, the following are equivalent.
(i)
φ is norm attainable(ii)
Bothφ+ and φ− are norm attainable(iii)
φ+ or φ− is norm oneProof.
i⇒ii. Choose x∈BΩ such that φx=∥φ∥=1. From
(1)1=∥φ∥=φx=φ+x++φ−x−−φ+x−−φ−x+=∥φ+∥∥x+∥+∥φ−∥∥x−∥−φ+x−−φ−x+=∥φ∥∥x+∥+∥x−∥=∥φ∥∥x∥=∥φ∥=1,
we obtainφ+x+=∥φ+∥∥x+∥;φ−x−=∥φ−∥∥x−∥ and φ+x−=φ−x+=0 since φ±x±≤∥φ±∥∥x±∥ and φ±,x± are nonnegative.
ii⇒iii. This follows obviously.
iii⇒i. We suppose that φ+ is norm one and norm attainable. Choose x∈BΩ+ such that φ+x=∥φ∥=1. We have φ−x=0. Indeed, 1=∥φ∥≥∣φ∣∣x∣≥φ+x+φ−∣x∣=1+φ−∣x∣≥1, which implies that φ−x+=φ−x−=0. Hence, φx=φ+x=∥φ∥=1.Theorem 19.
LetΩ∈AL and θ≤φ∈BΩ∗. Then, the following are equivalent.
(i)
φ is norm attainable(ii)
There existsθ≤x≠θ such that φy=∥y∥forallE¯Ω, the norm closure of EΩ where EΩ=y∈Ω:θ≤y≤nx,forsomen>0(iii)
There existsθ≠x∈Ω+ such that among BΩ∗=ψ∈Ω∗;∥ψ∥=1, φ is maximal on E¯ΩProof.
i⇒ii. Choose x∈BΩ satisfying φx=∥x∥=1. We have x∈θ for 1=φx=φx+=φx+−φ−≤φ+≤∥x+∥−∥x−∥≤1 by Lemma 18 which implies ∥x−∥=0. Now, if we consider y∈E¯Ω, we need to prove that φy=∥y∥. Since φ is continuous, let y∈E¯Ω, i.e., θ≤y≤nx for some n≥0. But since n=φnx=φnx−y+φy≤∥nx−y∥+∥y∥=∥nx∥=n, we have φy=∥y∥.ii⇒iii. Follows trivially from ii⇒iii in Lemma 18 and abstractness of AL.iii⇒i. Let φ be maximal in BΩ∗ on E¯Ω for some θ≠x∈Ω+. Choose ψ∈BΩ∗ such that ψx=∥x∥, then by φx≥ψx=∥x∥, it is clear that φ is norm attainable at x/∥x∥.Proposition 20.
LetΩ∈AM be σ-complete and φ∈Ω∗. Then, for any ε>0, there exists a subspace E of Ω=E+E⊥ and ∥φ+E⊥∥<ε,∥φ−E∥<ε.Proof.
Choosex∈DΩ with the property that φ+x>∥φ∥−ε, and E=x⊥⊥. Then, x+∈E,x−∈E⊥ and Ω=E+E⊥. Furthermore, for x∈DΩ, we have that ∥φ+E∥+∥φ+E⊥∥+∥φ−E∥+∥φ−E⊥∥=∥φ+∣∥+∥φ+∥=∥φ∥<φx+ε=φ+Ex+φ+E⊥x−φ−Ex−φ+E⊥x+ε because φ+E⊥x≤0 and φ−Ex≥0. So, we conclude that ∥φ+E⊥∥+∥φ−E∥=∥φ+∥−∥φ+E∥+∥φ−∥−∥φ−E∥≤∥φ+∥−φ+Ex+∥φ−∥−φ−E⊥x<φ+E⊥x−φ−Ex+ε≤ε. This completes the proof as required.Theorem 21.
Let a Banach latticeΩ be bounded σ-complete and BΩ order-closed, then every positive bounded linear φ∈Ω∗ is norm attainable.Proof.
Considerxn≥0∈DΩ such that φxn⟶∥φ∥. Since Ω is bounded σ-complete and BΩ is closed under order, y=∨nxn exists in Ω and ∥y∥=1. Hence, y≥xn≥0 and φ≥0 implies ∥φ∥≥φy≥φxn⟶∥φ∥. So, x∈DΩ exists which satisfies the norm-attainability condition, φx=∥φ∥, for functionals and this completes the proof.Theorem 22.
LetΩ∈AM be bounded σ-complete and BΩ order-closed, then φ∈Ω∗ is norm attainable if and only if there exists a subspace E of Ω such that φ+=φE,φ−=−φE⊥.Proof.
Necessity. Let x∈BΩ be having the property that φ∣x=∥φ∥, and define E=x−⊥. Then, Ω=E+E⊥ and x+∈E,x−∈E⊥. Now, ∥φ∥=∥φE∥+∥φE⊥∥; we need to show that φ+=φE and φ−=−φE⊥, it is enough that φE≥0 and −φE⊥≥0. Indeed, if φEy<0 for some y≥0∈DΩ then we let y∈E. So, z=−x−−y satisfies ∥z∥=max∥x∥,∥y∥=1, and hence, ∥φ−∥=φ−−z=φz−φ+z≥φz=φE⊥−x−−φEy>φE⊥−x−=−φE⊥−x. Now since ∥φ+∥≥φxE=φEx, this is contrary to ∥φ∥=∥φ+∥+∥φ−∥>φEx−φE⊥x=φx=∥φ∥. Also, −φE⊥≥0 follows analogously.
Sufficiency. We know that there exists x,y≥0∈DΩ such that φ+x=∥φ+∥ and φ−x=∥φ−∥ from Theorem 21. Since φ+=φE and φ−=−φE⊥, we let x∈E and y∈E⊥ and u∈x−y. Then, u=∥x−y∥=max∥x∥,∥y∥=1 and so we have ∥φ∥=∥φ+∥+∥φ−∥=φ+x+φ−y=φEx+φE⊥−y=φu.Corollary 23.
LetΩ∈AM be a σ-complete and φ∈DΩ∗. Then, φ∈extBΩ∗ if and only if φxφy=0 for all x,y∈Ω satisfying x∧y=0.Proof.
Necessity. If there exists x,y∈Ω satisfying x∧y=0 but φx>0 and φy>0, then we set E=y⊥, and Ω=E+E⊥. Let ψ=φE and τ=φE⊥. Then, ∥ψ∥>0,∥ψ∥>0 since x∈E,y∈E⊥. Therefore, φ=∥ψ∥ψ/∥ψ∥+∥τ∥τ/∥τ∥ and ∥ψ∥+∥τ∥=∥φ∥=1. Hence, φ∈extBΩ.
Sufficiency. First, we show ∥φ+∥∥φ−∥=0. In fact, for any ε>0, by Theorem 21, there exists two orthogonal subspaces E,F∈Ω such that Ω=E+F and ∥φ−E∥<ε,∥φ+F∥<ε. Choose x∈BΩ such that φx>∥φ∥−ε, and let x=u+v, where u∈E and v∈F. Then, φuφv=0 since u∧v=0. If φv=0 then ∥φ∥−ε<φx=φ+Eu−φ−Eu≤∥φ+E∥+∥φ−E∥<∥φ+∥+ε. Let ε⟶0, we find ∥φ−∥=∥φ∥−∥φ+∥=0. Similarly, if φu=0, then ∥φ+∥=0. Hence, without loss of generality, we assume φ=φ+. Let ψ,τ∈DΩ∗ satisfy 2φ=ψ+τ. Then, 2φ=ψ++τ+−ψ−+τ−, and hence, ∥2φ∥=∥ψ+∥+∥τ+∥+∥ψ−∥+∥τ−∥=∥ψ∥+∥τ∥=2=∥2φ∥. Thus, ψ++τ+=2φ and ψ−=τ−=0. Now, we show ψ=τ=φ, that is, φ∈extBΩ∗. This follows if we prove that ψy=τy=0 whenever φy=0; this means φ=aψ=bτ, but φ,ψ,τ∈DΩ∗ and 2φ=ψ+τ, so a=b=1, this means assume y≥0; then from ψy≥0,τy≥0 and ψy+τy=2φy=0. We have ψy=τy=0. For the general case, since φy=0 and by the condition given in theorem φy+φy−=0, we have φy+=φy−=0; hence, the condition ψy=τy=0 follows from the first case.


## 4. Norm-Attainability for Nonpower Operators

This section deals with norm-attainability for usual Hilbert space operators. To avoid ambiguity and without loss of generality, we use the term “nonpower” to differentiate these operators from those in the next section. The first result is a proof of norm-attainability condition for compact self adjoint operators in a dense separable infinite dimensional complex Hilbert space.Theorem 24.
LetH be a dense separable infinite dimensional complex Hilbert space. Let T:H⟶H be a compact and self-adjoint operator on Hilbert space H. Let D be unit sphere in H. Then, there exists a vector x0∈D such that ∥Tx0∥=∥T∥.Proof.
From an analogy of the definition of the usual operator norm, we know that∥T∥=supx∈D∥Tx∥, so there exists a sequence of elements x1,x2,⋯∈D such that limn→∞∥Txn∥=∥T∥. Since T let y0=limn→∞TxnexistsinH. Let Y=Spanx1,x2,⋯¯. Then, it is a closed subspace of H, hence a Hilbert space. So, Y is reflexive and separable. Thus, there exists a subsequence of xn, say xj1,xj2,xj3,⋯ with 1≤j1<j3<⋯, such that xjk converge weakly in Y to some point x0∈Y. Now, for each z∈Y, Tx0,z=x0,Tx=limk→∞xjk,Tz=limk→Txxk,z=y0,z. Now, the second equality holds since xjk converge weakly to x0. Indeed, x↦x,Tz is a bounded linear functional on Y, and the last equality holds because we know from the fact that y0=limn→∞TxnexistsinH. So, limk→∞Txjk=y0 exists and .,. is continuous. Since Tx0,z=x0,Tx=limk→∞xjk,Tz=limk→Txxk,z=y0,z holds for z∈Y, we conclude that Tx0=y0. Hence, ∥Tx0∥=∥y0∥=∥limk→∞Txn∥=limk→∞∥Txn∥=∥T∥. Now, since xjk converge weakly to x0, we have x0,x0=limk→∞xjk,x0 and ∣xjk,x0∣≤∥xjk∥.∥x0∥=1 for all k, hence ∥x0∥≤1. We cannot have ∥x0∥<1 since then ∥Tx0∥≤∥T∥.∥x0∥<∥T∥ which is a contradiction. Hence, ∥x0∥=1, i.e., x0∈D. Hence, the existence of x0 is proved as required.Corollary 25.
LetH be a dense separable infinite dimensional complex Hilbert space. Let T:H⟶H be a compact and self-adjoint operator on Hilbert space H and let Y⊂H be a subspace such that TY⊂Y. Let TY⊥⊂Y⊥ be a subspace such that TY⊂Y. Then, TY⊥⊂Y⊥, and TY⊥:Y⊥⟶Y⊥ is a bounded self-adjoint linear operator on Hilbert space Y⊥, with the norm ∥TY⊥∥≤∥T∥.Proof.
Choosez∈Y⊥. Now, for each y∈Y,Tz,y=z,Ty=0 since Ty∈TY⊂Y and z∈Y⊥. Clearly, Tz∈Y⊥. Hence, TY⊥⊂Y⊥.The next proposition in this section is a very important refined result for a characterization of operators inBH whose analogies can be found in [2] Theorem 2.1 or [4] Proposition 2.1.Lemma 26.
LetH be a reflexive, dense, separable, infinite dimensional complex Hilbert space. Let S∈BH,β∈W0S, and α>0 be given. Then, there exists a self-adjoint compact operator Z∈BH such that ∥S∥=∥Z∥, with ∥S−Z∥<α. Furthermore, there exists a unit vector η∈H such that ∥Zη∥=∥Z∥ with Zη,η=β.Corollary 27.
LetS,T∈BH. If both S and T are norm attainable, then the basic elementary operator MS,T is also norm attainable.Proof.
For any pairS,T, it is known that ∥MS,T∥=∥S∥∥T∥. We can assume that ∥S∥=∥T∥=1. If both S and T are norm attainable, then there exists unit vectors ζ and η with ∥Sζ∥=∥Tη∥=1. We can therefore define an operator X by X=·,Tηζ. Clearly, ∥X∥=1. Therefore, we have ∥SXT∥≥∥SXTη∥=∥∥Tη∥2Sζ∥=1. Hence, ∥MS,TX∥=∥SXT∥=1, that is, MS,T is also norm attainable.Proposition 28.
If the elementary operatorTAi,Bi is norm attainable, then there exists an isometry or a coisometry Q0 such that ∥TAi,Bi∥=∥Σi=1nAiQ0Bi∥.Proof.
From [7], if the elementary operator TAi,Bi is norm attainable, then there exists a contractive operator X0∈BH such that ∥TAi,Bi∥=∥Σi=1nAiX0Bi∥. Lastly, it is sufficient enough to show that X0=1/2Γ1+Γ2 for two isometries or coisometries Γ1,Γ2∈BH. This follows immediately from Lemma 3.1 in [7].


## 5. New Notion of Norm-Attainability for Power Operators

In this section, we give new notions of norm-attainability for power operators. First, we begin with some auxiliary results.Proposition 29.
LetT∈NAH. Then, T is p-norm attainable if it is p-normal.Proof.
LetT∈NAH be p-normal, i.e., TpT∗=T∗Tp. By replacing T∗ with its power raised to p, it is easy to see that TpT∗p=T∗pTp. Hence, Tp is normal. Now, if we consider Tp to be normal, then TpT=TTp since T∗Tp=TpT∗ by Fuglede property. Therefore, T is p-normal. But T∈NAH and Tp is normal so it follows that there exists a unit vector x∈H such that ∥Tpx∥=∥Tp∥, for any p∈ℕ.Remark 30.
Every norm-attainable operator and every bounded normal operator isp-norm attainable and p-normal, respectively, for any p∈ℕ. However, the converses are not true in general.Theorem 31.
LetNApH be the set of all p-norm-attainable operators on H. Then, NApH is a closed subset of NAH which is closed under scalar multiplication if and only if for any T∈NAH,T is p-normal.Proof.
Considerπ∈K and let T be p-normal. By carrying out a simple calculation, we find that πTpπT∗=πT∗πTp and so πT is p-normal. If T∈NAH, then the converse follows immediately by taking limits over a sequence of vectors in H. Therefore, T is p-normal.Corollary 32.
LetT∈NAH be p-norm attainable. Then, the following conditions hold:
(i)
UTU∗ is p-norm attainable for a unitary operator U(ii)
T∗ is p-norm attainable(iii)
T−1 is p-norm attainable if it exists(iv)
If there exists a unitary equivalence betweenT0∈NAH and T, then T0 is p-norm attainable(v)
IfG is a unitarily invariant subspace of H such that G reduces T, then T0=T/G is a p-norm-attainable operatorProof.
i Since U is unitary then UU∗=U∗U=I, where I is the identity operator then by Definition 5 and by Theorem 31 we obtain the result. ii Since T is p-norm attainable from Proposition 29, Tp is p-norm attainable and so T∗p is p-norm attainable. Consequently, T∗ is p-norm attainable. iii Suppose that T−1 exists. Since T is p-norm attainable, Tp is p-norm attainable. Since Tp−1=T−1p is p-norm attainable, T−1 is a p-norm-attainable operator. iv Follows immediately from i.iv Follows from the fact that G is invariant under T. The rest is clear.Theorem 33.
LetA,B∈NApH be commuting p-norm-attainable operators, then AB is a p-norm-attainable operator.Proof.
Suppose thatA,B∈NApH. Since A,B are commuting p-norm-attainable operators, then Ap,Bp are commuting normal operator. From [3], ApBp is a p-norm-attainable operator and hence norm attainable. Lastly, ApBp=ABp=BAp is normal and norm attainable. Hence, AB is a p-norm-attainable operator.Remark 34.
Not allp-norm-attainable operators are p-normal. In fact, the following example shows that the sum of two commuting p-normal operators need not be p-normal.Example 35.
Consider(2)A=1001,B=0100.
Clearly,A and B are commuting 2-normal operators. Now,
(3)A+B=1101,A+B2=1201are not normal. So A+B is not 2-normal. Recall that B is self-adjoint.


## 6. Open Questions

In conclusion, we have characterized norm-attainability for functionals in Banach spaces. Moreover, we have given a new notion of norm-attainability for power operators and also for elementary operators in normed spaces. The following two open questions arise naturally:(1)
LetH be a reflexive, dense, separable, infinite dimensional complex Hilbert space. Does there exist a unit vector x0∈H for a bounded self-adjoint operator A:H⟶H such that ∥Ax0∥<∥A∥?(2)
When doesp-norm-attainablity and p-normality coincide in general Banach spaces?



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