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On Norm Estimates for Derivations in Norm-Attainable Classes

J. Z. Nyabonyi1,∗, N. B. Okelo2, R. K. Obogi1
1Department of Mathematics and Actuarial Science, Kisii University, Kenya

nyabonyijanes@yahoo.com, krbertobogi@yahoo.com
2Department of Pure and Applied Mathematics,

Jaramogi Oginga Odinga University of Science and Technology, Kenya
bnyaare@yahoo.com

∗Correspondence: nyabonyijanes@yahoo.com

Abstract. In this note, we provide detailed characterization of operators in terms of norm-attainabilityand norm estimates in Banach algebras. In particular, we establish the necessary and sufficientconditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes.

1. Introduction
The norm of a derivation was first introduced by Stampfli [49], who determined the inner derivation

δT0 : A0 → T0A0 − A0T0 which acts on B(H), the algebra of all bounded linear operators on acomplex Hilbert space H. Further, ‖δT0‖ = inf 2‖T0 −λI0‖, for every complex λ was shown. Fora normal operator T , ‖δT0‖ can be expressed as the geometry of the spectrum of T0. Johnson [21]established methods which apply to a uniformly convex spaces with a large class, i.e the formula
‖δT‖ is false in lp and Lp(0, 1) 1 < p < ∞, p 6= 2. For L1 space the formula is true for areal case and not for a complex case whose space dimension is 3 or more. Johnson [20] foundthat a derivation on B(H) is a mapping ∆ : B(H) → B(H) with ∆(AS) = A∆(S) + ∆(A)S, where
A,S ∈ B(H). Such derivations are necessarily continuous and if S ∈ B(H) then ∆S(A) = AS−SAis a derivation on B(H). Gajendragadka [18] was concerned with the Von Neumann algebra andcomputed the norm of a derivation. Specifically, it was proved that the Von Neumann algebra actson a separable Hilbert space H, whereby if T is in U and δT is the derivation induced by T, then
‖δT |U‖ = 2 inf ‖T −Z‖, where Z is the centre of U. Therefore, Anderson [3] in his investigation onnormal derivations with the operators A,C ∈ B(H) proved if A is normal and AC commute, for every
X ∈ B(H), ‖δA(X) + C‖≥‖C‖. Therefore, the inequality showed that the kernel and the range of
δA are orthogonal to δA which is the commutation of {A}′ of A. Kyle [24] examined the relationship

Received: 22 Jun 2022.
Key words and phrases. derivation; norm; norm-attainability; Banach algebra.1

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of the numerical range of inner derivation and that of the implementing element. Kyle [25] studiednorms of inner derivations and used their properties and concluded that a closed subset of allderivations on a C∗-algebra, forms the set of inner derivations and obtained the result which is aconverse of Stampfli [49]. Charles and Steve [11] answered the question when X = T by structurecharacterization of compact derivations of C∗-algebras. Moreover, the structure of weak compactderivations of C∗-algebras was determined and as immediate corollaries of these results, conditionsthat were necessary and sufficient were obtained so that C∗-algebras can admit a non-zero compactor weakly compact derivation. Stampfli [50] studied operators on Hilbert spaces and their propertiesinducing a derivation whose closure is self-adjoint after the range of such operators are termed
D-symmetric and then characterized compact D-symmetric operators. Erik [16] established thatany operator T on a Hilbert space H with a cyclic vector has a property with a finite spectrum.Mecheri [31] established that T (X) is linear for any m-linear derivation and hence, the topologyof Von Neumann algebra X of type I is automatically continuous in measure with center m and thesemi-finite trace τ which is normal is faithful. Therefore, T (X) is the algebra of all τ-measurableoperators affiliated with X. Mathieu [29] proved that for non-zero derivations, the product of twoprime C∗-algebras are bounded if both of them are bounded. In [51], two automatic continuityproblems for derivations on commutating Banach algebras were discussed, that is, derivation on acommutative algebra is mapped onto the radical, and Banach algebras are continuous on semiprimederivations. Bresar, Zalar [9] showed that a Jordan ∗-derivation is the map δa(x) = ax − x∗a forfixed a ∈ U; hence, the derivation is inner. Douglas [15] continued the study of Ws(Y ) which wasconsiderably more amenable where Archbold [1] defined the smallest numbers to be [0,∞] andintroduced two constants W (Y ) and Wt(Y ) such that d(y,Z(Y )) ≤ W (Y )‖D(y,Y )‖, for all y ∈ Yand d(y,Z(Y )) ≤ Ws(Y )‖D(y,Y )‖, for all y = y∗ ∈ Y. The author in [26] showed that for the nthorder commutator [[[k(B),Y ],Y ], ...,Y ], a formula was obtained in terms of the Frechet derivatives
Smk(B) in which the formula illustrated was used to obtain bounds for norms of a generalizedcommutator k(B)Y −Y k(B) and their higher order analogues. In [17], numerical ranges of 2 x 2matrices were determined and the convex of the numerical range for any Hilbert space operator wasestablished in Toeplitz-Hausdorff theorem and the relation of the numerical range to that of spectrumwas discussed. Further, the closure of the numerical range is contained in the spectrum and theintersection of closures of the numerical range of all operators were asserted by Hildebrandt’stheorem. Considering results on special cases [10], established that ‖PXQ + QXP‖ ≥ ‖P‖‖Q‖.Chi-Kwong [13] established that for an n x n matrix X, the numerical range W (X) has manyproperties which can be used to locate eigenvalues to obtain norm bounds. Algebraic and analyticproperties were deduced which help in finding the dilations of simple structures. Let the linearoperators Xi and Yi , 1 ≤ i ≤ n act on separate Hilbert space H, therefore, Hong-Ke, Yue-qing [19]proved that sup{‖∑ni=1PiXQi‖ : X ∈ B(H),‖X‖ ≤ 1} = sup{‖∑ni=1PiTQi‖ : UU∗ = T∗U =

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I,U ∈ B(H)}. In addition, Okelo, Agure and Ambogo [35] established the norm of Jordan elementaryoperator UA,B : B(H) → B(H) which is given by UA,B = AY B + BY A, ∀Y ∈ B(H) and A,B fixedin B(H) and showed that ‖UA,B ‖≥‖A‖‖B‖ and then characterized the norm-attainable operatorsusing this norm. Inner derivations implemented by norm-attainable elements of a C∗-algebra hasrelation to those of ideals and primitive ideals. Since there is a relationship between the constants
A(ξ) and Asξ of C∗-algebras to the ideals and primitive ideals then related results have beengiven in general Banch settins. Okelo, Agure and Oleche [38] gave results on necessary andsufficient conditions for norm-attainable operators and also studied norm-attainable operators andgeneralized derivations. Okelo [37] extended the work by presenting new results on conditionsthat are necessary and sufficient for norm-attainability for operators in Hilbert space, elementaryoperators and generalized derivations. Further, Okelo [37] established that a unit vector exists
λ ∈ H, ‖λ‖ = 1 such that ‖Sλ‖ = ‖S‖ with 〈Sλ,λ〉 = η. Results from [23] showed that every Jordanderivation of the trivial extension of A by M, under certain conditions, is the sum of a derivationand antiderivation. In [10], the author studied norm-attainable operators that are convergent andestablished norm-attainability of operators via projective tensor norm. Wickstead [52] showed thatif an atomic Banach lattice Z with a continuous norm order, X,Y ∈ T r and MX,Y is the operatoron T r (Z) defined by MX,Y (A) = XAY, then ‖MX,Y‖r = ‖X‖r‖Y‖r but there is no real β > 0such that ‖MX,Y‖r = β‖X‖r‖Y‖r. Okelo [36] outlined the theory of normal, self-adjoint and norm-attainable operators then presented norms of operators in Hilbert spaces. In [8] the author provedthat for a linear map ∆ : U → U, ∆(XY ) = ∆(X)Y + ∆X(Y ) for each X,Y ∈ U is a derivation,and for any two derivations ∆ and ∆′ on a C∗-algebra U there exists a derivation δ ∈ U suchthat ∆∆′ = δ2 if and only if either ∆′ = 0 or ∆ = f ∆′ for any f ∈ C. Clifford [12] studiedhypercyclic generalized derivations acting on separable ideals of operators and also identifiedconcrete examples and established some conditions that are necessary and sufficient for theirhypercyclicity. Okelo [36] considered orthogonal and norm-attainable operators in Banach spaces,gave in details the characterization and generalizations of norm-attainability and orthogonality.The conditions that are sufficient and necessary for norm-attainability of operators on a Hilbertspace, the result on orthogonal range and the kernel of elementary operators implemented by norm-attainable operators in Banach spaces were also given. Okelo [34] characterized norm-attainableclasses in terms of orthogonality by giving norm-attainability conditions that were necessary andsufficient for Hilbert space operators first and the orthogonality result on the range and kernel ofelementary operators when implemented by norm-attainable operators in norm-attainable classeswere also given. Okelo [38] gave conditions for norm-attainability for linear functionals in Banachspaces, non-power operators on H and elementary operators and also gave a new notion of norm-attainability for power operators then characterized norm-attainable operators in normed spaces.In [51] determined the norm of the inner Jordan ∗-derivation δS : X → SX − X∗S acting on the

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Banach algebra B(H). It was shown that ‖δS‖≥ 2 supλ∈W0(S) |=λ| in which W0(S) is the maximalnumerical range of operator S. The work of [1] obtained precisely when zero belongs to maximalnumerical range of composition operators on H and then characterized the norm-attainability ofderivations on B(H). In Okelo [41] norm-attainability for hyponormal operators that are compactwere characterized, sufficient conditions for a compact hyponormal operator that is linear andbounded on an infinite dimension for a complex Hilbert space to be norm attainable were given.Further, the structure and other properties of compact hyponormal operators when they are self-adjoint, normal and norm attainable with their commutators were discussed in general. Lumer [27]obtained a sharp estimate not only from |sp(R)| equal to spectral radius of R but indeed for |sp(R)|in terms of sup(|X(R)|, |X(Rn)|1/n),n being any positive even integer. In [18] the author studiedthe algebra of functions that are continuous on [0, 1] and are ‖.‖w -approximate polynomial; i.epoint-wise functions of limits of ‖.‖w -Cauchy sequence of polynomial. Archbold [1] investigatedwhether the simple triangle inequality ‖T (a,A)‖ ≤ 2t(a,Z) if applied holds. D(A) was definedto be a minimum value D in [0,∞] such that t(a,Z) ≤ D‖T (a,A)‖. The behaviour of D in idealsand quotients were discussed which proved that Ds(A) ≤ 1 for a weakly central C∗-algebra A andconsidered a class of n-homogeneous C∗-algebras that are special. D and Ds were investigatedand approximated finite-dimension (AF )C∗-algebra in that context and an example was given toshow certain estimates. The results of [44] showed that for a certain Von Neumann algebra U,a constant F existed such that dist(T,U) ≤ F supP∈latU ‖P⊥TP‖∀T ∈ B(H). The work wasextended to a Von Neumann algebra U and showed that there exists a constant G ∈ B(H),dist(T,U) ≤ G‖∆T |U′‖ where δT is the derivation δT (S) = ST − TS thus proving that theinequality holds for large classes of Von Neumann algebras. In [14] the researcher considered
λ(M) defined as the smallest number ‖Z‖2 of Z that satisfy [Z∗,Z] = M and showed that
1 ≤ λ(M) ≤ 2. Matej [28] estimated the distance of d1 and d2 to the generalized derivations andthe normed algebra of P and considered the cases when P is an ultraprime, when d1 = d2 and
P are ultrasemiprime and when P is a Von Neumann algebra we have the equation ‖P + Q‖ =
‖P‖ + ‖Q‖, P,Q ∈ B(H). Further, a constructive proof was provided that a minimum bound isnot valid and a relevant method to analyze the problem on estimation of eigenvalues such aninterpolation matrix was commented on. The norm property was done by Cabrera, Rodriguez [10]for basic elementary operators and obtained ‖Ma,b‖ ≤ 2‖a‖‖b‖, for Jordan elementary operator
‖U‖ = ‖Ma,b‖ + ‖Ma,b‖, ‖Ma,b‖ + ‖Ma,b‖ ≤ 2‖a‖‖b‖ for the upper estimates. In fact, [30] gavean estimate on matrix-valued function that is regular and showed that for normal matrices it isattainable and investigated their stability. Kittaneh [26] established the orthogonality, kerneland the range of a normal derivation associated with norm ideals of operators with respect tothe unitarily invariant norms. Results related to orthorgonality of some derivation that are notnormal were also obtained. Stacho and Zalar [48] established the lower estimates for elementary

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operators of Jordan type in standard Banach algebras. Danko [14] established that for all unitarilyinvariant norms and for bounded Hilbert space operators there exist {xn}n ⊆ H which is a unitsequence such that limn‖C −ω‖xn = 0. From [11], ‖A‖ ∈ σ(A) if and only if ‖A‖ ∈ σap(A) also
σ(A) ⊆ W (A) (spectral inclusion) and if ω(A) = ‖A‖, then γ(A) = ‖A‖. Therefore, the resultimplied that ‖A‖ ⊆ W (A) if and only if ‖A‖ ∈ σ(A). In fact, Megginson [32] established that forall Y ∈ K, then δB(Y ) ∈ J and ‖BY − Y B‖K = ‖(B − λ)Y − Y (B − α)‖J ≤ 2‖B − α‖‖Y‖Kfor all α ∈ C. Hence, ‖δB(Y )‖K ≤ 2d(B)‖Y‖K, implying that ‖δB|K‖ ≤ 2d(B). Further, thenotion of R-universal operators was introduced and that R-universal is an operator A ∈ B(H) if
‖δB|K‖ = 2d(B) for every norm ideal K ∈ B(H). Landsman [23] proved that for a standard operatoralgebra on H ‖Ma,b‖+‖Ma,b‖≥ 2(√2−1)‖a‖‖b‖. Therefore, both the lower norm and upper normbounds have been established for normally represented elementary operators. The work of [3] had anestimate on transfer functions of stable linear time-invariant systems on stochastic assumptions. Theapproach of nonparametric minimax was adopted to measure the estimate accurately, an estimator ofquality was measured over a family of transfer functions by its worst case error. In [32] the authorestablished that for a holomorphic functions f with Re{gf ′(g)} > α and Re{gf ′′(g)/f ′(g)} >
α−1, (0 ≤ α < 1) respectively in {|g| < 1}, estimates of sup|g|<1(1−|g|2)|f ′′(g)/f ′(g)| were givenand functions Gelfer-convex of exponential order α,β was also considered. Milos, Dragoljub [33]considered elementary operators x → ∑nj=1vjxwj that acts on a Banach algebra. The ascentestimation and lower bound estimation of an operator was given. Barraa and Boumazgour [4]showed that the norm of bounded operators more than one on a Hilbert space is the same asthe sum of the norms and showed that δS,A,B is convexoid with the convex hull of its spectrumif and only if A and B are convexoid. Richard [44] established the CB-norms of elementaryoperators and the lower bounds for norms on B(H). The result was concerned with the operator
UA,BX = AXB+BXA which showed that ‖UA,B‖≥‖A‖‖B‖ which proved a conjecture of Mathieu,other results and formula of ‖UA,B‖CB and ‖UA,B‖ were established. Richard [45] provided theHaagerup estimation on the norm of elementary operators that are completely bounded. Seddik [46]proved that lower estimate bound ‖TM,N‖ ≥ 2(√2 − 1)‖M‖‖N‖ holds, if it is either a standardoperator algebra or a norm ideal on B(H) and M,N ∈ B(H). Florin, Alexandra [17] estimatedthe norm of operator Hθ,λ = Uθ + U∗θ + (λ/2)(Vθ + V ∗θ ) which is an element on a C∗-algebra
Aθ = C

∗(Uθ,Vθ unitaries : UθVθ = e2πiθVθUθ), and proved that for every λ ∈C and θ ∈ [14, 12] theinequality ‖Hθ,λ‖≤ √4 + λ2 − (1 − 1tanθ,λ)(1 − √1+cos24πθ2 )min{4,λ2} holds. This significantlyimproved the inequality ‖Hθ,2‖≤ 2√2,θ ∈ [14, 12], conjectured by [18]. The author in [31] consideredcommuting matrices of matrix valued analytic function and established a norm estimate, in particular,two matrices of matrix valued functions on a tensor product in a Euclidean space were explored. In [5]the research communicated results on complex symmetric operator theory and showed that two non-trivial examples were of great use in studying Schrödinger operators. The work of [43] showed that

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triangle inequality served an upper norm bound for the sum operators that is sup{‖T∗RT +V ∗SV‖ :
TandV} are unitaries. The result discussed had relationship to normal dilations, spectral setsand the Von Neumann inequality. Yong, Toshiyuki [53] gave a norm estimate on pre-Schwarzianderivatives of a specific type of convex functions by introducing a maximal operator of independentinterest of a given kind. The relationship between the convex functions and the Hardy spaces wasdiscussed. In [16] the author analyzed the structure of the set D = {y ∈ D(δ) : limn→∞ ∆n(y) =
∆(y)} for convergence of the generators that are pointwise where α is an approximate innerflow on a C∗-algebra T with generator ∆ and ∆n for bounded generators of the approximateflows αn. In fact, the relationship of D and various cores related to spectral subspaces wereexamined. Seddik [47] showed that Q is a normal operator which is invertible in B(H) if theestimate ‖Q⊗Q−1 + Q−1 ⊗Q‖λ ≤ ‖Q‖‖Q−1‖ + 1‖Q‖‖Q−1‖ holds, such that ‖.‖λ is the injectivenorm on the tensor product B(H) ⊗ B(H), when Q is invertible self-adjoint then the equationbecomes an equality. Bonyo and Agure [7] characterized the norm of inner derivation on normideal to be equal to the quotient algebra and investigated them when they are implemented bynormal and hyponormal operators on norm ideals. A hyponormal X is a bounded linear operatoron a Hilbert space H if X∗X − XX∗ ≥ 0 and is normal if X∗X = XX∗. Bonyo and Agure [8]investigated the relation of the diameter of the numerical range of an operator B ∈ B(H) and thenorm of inner derivation implemented by B on a norm ideal J and considered the application of
S-universality to the relation. Bonyo and Agure [6] defined inner derivations implemented by A,Brespectively on B(H) by δA(Y ) = AY − Y A, δB(Y ) = BY − Y B and generalized derivation by
δA,B (Y ) = AY −Y B ∀ Y ∈ B(H). Further, a relationship between the norms of δA,δB and δA,Bon B(H) was established, specifically when the operators A,B are S-universal. Ber, Sukochev [5]showed that for every self-adjoint element b ∈ S(N) a scalar λ0 ∈ R exists such that ∀ ε > 0,then there exists a unital element uε from N satisfy |[b,uε]| ≥ (1 −ε)|b−λ01|. From this result aconsequence is that for any derivation δ on N with the range on an ideal I ⊆ N the derivation δis inner i.e δ(.) = δa(.) = [a,.] and a ∈ I. Pablo, Jussi, Mikael [42] provided theoretic estimate oftwo functions for the essential norm as a composition operator Cϕ that acts on the space BMOA;one in terms of the n-th power ϕn denoted by ϕ and the other involved the Nevanlinna countingfunction. The research of [20] introduced a new type of norm for semimartangles, the defined norm ofquasimartangales and then characterized the square integrable semimartangales. In [4] the authorgave the result on lower bound of the norms for finite dimensional operators. The work of [14]determined the norm of two-sided symmetric operator in an algebra. More precisely, the lowerbound of the operator using injective tensor norm was investigated. Further, the inner derivationnorm on irreducible C∗-algebra was determined and Stampfli’s [49] result for these algebras wasconfirmed.

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2. Preliminaries

This section provides the basic concepts which are useful in the sequel.
Definition 1 ( [1], Definition 1.5). A Banach ∗-algebra T is called C∗-algebra if ‖tt∗‖ =
‖t‖2, ∀ t ∈T .

Definition 2 ( [37], Definition 2.1). Elementary operator T : B(H) → B(H) is defined by
TDi,Ei (X) =

∑n
i=1Di X Ei ∀ X ∈ B(H) and ∀ Di,Ei fixed in B(H) where i = 1, ...,n. For

B(H), we define the particular elementary operators as below:

(i). Left multiplication operator LD : B(H) → B(H) by LD(X) = DX, ∀ X ∈ B(H).(ii). Right multiplication operator RE : B(H) → B(H) by RE(X) = XE, ∀ X ∈ B(H).(iii). Generalized derivation (implemented by D,E) by δD,E = LD −RE.(iv). Inner derivation (implemented by D) by δD(X) = DX −XD.(v). Basic elementary operator (implemented by D,E) by MD,E(X) = DXE, ∀ X ∈ B(H).(vi). Jordan elementary operator (implemented by D,E) by UD,E(X) = DXE + EXD, ∀ X ∈
B(H).

Definition 3 ( [49], Definition 2.3). A derivation is a map D : U → U satisfying D(f g) = f D(g) +
D(f )g for all f ,g ∈ U.

Definition 4 ( [39], Definition 1.2). The maximal numerical range of an operator S is defined by:
W0(S) = {β : 〈St,t〉→ β, where ‖t‖ = 1 and ‖St‖→‖S‖}.

Definition 5 ( [35], Definition 2.1). An operator K is norm-attainable if t ∈ H exists which is a unit
vector such that ‖Kt‖ = ‖K‖. Moreover, it is self-adjoint if K = K∗.

3. Main Results
In this section, we give results on norm-attainability conditions an norm estimates for derivations.We begin with the following proposition.

Proposition 6. Let H be a complex Hilbert space and B(H) the algebra of all bounded linear
operators on H. A ∈ B(H) is norm-attainable if and only if its adjoint A∗ ∈ B(H) is norm-
attainable.

Proof. Given A ∈ B(H) is norm-attainable then we need to show that A∗ ∈ B(H) is norm-attainable. If A ∈ B(H) is norm-attainable then by definition of norm-attainability there exists aunit vector x ∈ H with ‖x‖ = 1 such that ‖Ax‖ = ‖A‖. That is, ‖AA∗x‖ = ‖A2x‖. Let η = Ax‖A‖,then η is a unit vector such that ‖η‖ = 1 this implies that ‖A∗η‖ = ‖A‖ = ‖A∗‖. Hence, A∗ isnorm-attainable. �

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The next result gives norm-attainability conditions for operators via the essential numerical range.An analogy of the same can be found in [37].
Proposition 7. Let A ∈ B(H), λ ∈ Wess(A) and η > 0. Then there exists A0 ∈ B(H) such that
‖A‖ = ‖A0‖ with ‖A−A0‖ > η.

Proof. See [37] for the proof. �
Remark 8. The set of all norm-attainable operators is denoted by NA(H), the set of all norm-
attainable self adjoint operators is denoted by NA∗(H) and the set of all norm-attainable elemen-
tary operators is denoted by ENA[B(H)].

At this point, we consider norm-attainability in a general set up. We begin with the followingproposition.
Proposition 9. Let D be the unit disc of a complex Hilbert space H and A : H → H be compact
and self adjoint. Then there exists x ∈ D such that ‖Ax‖ = ‖A‖.

Proof. By the definition of usual norm, we have ‖A‖ = supx∈D ‖Ax‖. So, there exists a sequence
x1,x2, ...,xn in D such that ‖Axn‖ = ‖A‖. But A is compact so let y0 = limn→∞Axn exist in
H. Suppose Y = span{x1,x2}, then it is a closed subspace of H. If we pick a subsequence xnkof xn, then it converges weakly to x and we have done 〈x,x〉 = limk→∞〈xnk,x〉 and |〈xnk,x〉| ≤
‖xnk‖‖x‖ = 1 for all k. Therefore, ‖x‖ ≤ 1 but we cannot have ‖x‖ < 1 since then ‖Ax‖ =
‖A‖‖x‖ < ‖T‖ which is a contradiction. Thus, ‖x‖ = 1 i.e x ∈ D. Hence, the existence of x isshown and thus completes the proof. �
At this point, we consider q-normality and q-norm-attainability.
Lemma 10. Let A ∈ NA(H) then A is q-norm-attainable if it is q-normal.

Proof. Let A ∈ NA(H) be q-normal i.e AqA∗ = A∗Aq. Raising A∗ to power q and using it toreplace A∗ we have Aq(A∗)q = (A∗)qAq. This shows that Aq is normal. Now AqA∗ = A∗Aq byFuglede property. Therefore, A is q-normal. However, A ∈ NA(H) and Aq is normal so it followsthat there exists a unit vector x ∈ H such that ‖Aqx‖ = ‖Aq‖, for any q ∈ N. Hence, Aq isnorm-attainable. �
Remark 11. Every norm-attainable operator and every self adjoint operator is q-norm-attainable
and q-normal for any q ∈N. However, the converse need not be true in general see [66].

Lemma 12. Let NAq(H) be the set of all q-norm-attainable operators on H. Then NAq(H)is a
closed subset of NA(H) which is algebraic if and only if for any A ∈ NA(H), A is q-normal.

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Proof. Let A be q-normal and pick λ ∈ K. By premultiplying by λ and postmultiplying by q asa power on the normal A we have (λA)q(λA)∗ = (λA)∗(λA)q. This proves the normality of λA.Now if A ∈ NA(H) then the converse is true if we take limits over a sequence of vectors in H andalso by Proposition 9. Therefore, A is a q-normal. �
Theorem 13. Let A ∈ NAq(H). Then the following conditions are true.

(i). A∗ is q-norm-attainable.(ii). V AV ∗ is q-normal, for a unitary operator V ∈ NAq(H).(iii). A−1 is q-norm-attainable if it exists.(iv). A0 = A/G is q-norm-attainable for some G which is a uniformly invariable subspace of Hwhich reduces to A.(v). A0 is uniformly equivalent to A implies A0 is norm-attainable.
Proof. (i). Since A ∈ NAq(H), then from Lemma 10, Aq is q-norm-attainable and so (A∗)q isnorm-attainable. Consequently, A∗ is q-norm-attainable.(ii). Since V is unitary then V V ∗ = V ∗V = I, where I is the identity operator. By definition ofnorm-attainability and Lemma 10 we obtain the desired results.(iii). If A−1 exists then since A is q-norm-attainable, Aq is q-norm-attainable. Now since A is

q-norm-attainable then by Lemma 10 Aq is q-norm-attainable. But (Aq)−1 = (A−1)q is
q-norm-attainable. So A−1 is q-norm-attainable.(iv). Follows from the fact that G invariant under A.(v). Follows from (iii) since V is unitary.

�

Corollary 14. Let Aq,Aq0 ∈ NAq(H) be commuting operators, then A,A0 ∈ NAq(H).

Proof. Since Aq,Aq0 ∈ NAq(H) are commuting then A,A0 are commuting normal operators. Bysupraposinormality of operators in dense classes we have A,A0 ∈ NAq(H) and hence are norm-attainable. Indeed, AqAq0 = (AA0)q = (A0A)q which is normal and norm-attainable. Hence,
A,A0 ∈ NAq(H). �

Remark 15. Not all q-norm-attainable operators are q-normal. Thus, the following example shows
that the two commuting q-normal operators need not be q-normal.

Example 16. Let A =
[

1 0

0 1

]
and A0 =

[
0 1

0 0

]
. Now A + A0 =

[
1 1

0 1

]
and (A + A0)2 =[

1 2

0 1

]
are not normal. So A + A0 is not 2-normal. We note that A0 is self-adjoint.

Lemma 17. The sum of norm-attainable operators is norm-attainable.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.9 10
Proof. Consider A,B ∈ B(H). We need to show that the sum of A and B is norm-attainable. For
A,B to be norm-attainable then there exists a unit vector x ∈ H such that ‖x‖ = 1, ‖(A + B)x‖ =
‖Ax +Bx‖ = ‖A+B‖ = ‖A‖+‖B‖. Since ‖Ax +Bx‖≤‖Ax‖+‖Bx‖≤‖A‖+‖Bx‖≤‖A‖+‖B‖then for an orthonormal sequence xn ∈ H we have limn→∞(‖Axn + Bxn‖) = ‖Ax + Bx‖. But since
A and B are norm-attainable we have ‖Ax +Bx‖ = ‖(A+B)x‖ = ‖A+B‖ is norm-attainable. �
Theorem 18. A norm-attainable operator perturbed by an identity operators is norm-attainable.

Proof. Let B ∈ B(H) be norm-attainable. Since B is norm-attainable then there exists a unitvector x0 ∈ H, an identity I ∈ B(H) and for every ε > 0 we have ‖(BI)x0‖ ≤ ‖BIx0‖ + ε ≤
‖B‖‖I‖‖x0‖ + ε. Since ε is arbitrary then it follows that ‖(BI)x0‖ ≤ ‖B‖‖I‖‖x0‖ = ‖B‖. Hence,
‖(BI)x0‖ = ‖B‖. �

At this point, we consider norm-attainability for elementary operators. We begin with inner deriva-tions.
Lemma 19. Let δA ∈ E[B(H)], then δA is norm-attainable if there exists a unit vector x0 ∈ H,
A ∈ NA(H) and 〈Ax0,x0〉 ∈ Wess(A).

Proof. For an operator A ∈ NA(H) we know that an operator is norm-attainable via essentialnumerical range from proposition 4.2. Now, we need to show that δA ∈E[B(H)] is norm-attainable.By the definition of inner derivation, δA = AY0−Y0A. Since A is norm-attainable then there exists aunit vector x0 ∈ H such that ‖x0‖ = 1, ‖Ax0‖ = ‖A‖. By orthogonality let y0 satisfy y0⊥{Ax0,x0}and a contractive Y0 be defined as a linear transformation Y0 : x0 → x0 with Ax0 → −Ax0 as
y0 → 0. Since Y0 is a bounded linear operator on H, then by norm-attainability ‖Y0x0‖ = ‖Y0‖ = 1and

‖AY0x0 −Y0Ax0‖ = ‖Ax0 − (−Ax0)‖ = 2‖A‖.

It follows from Lemma 3.1 in [49] that ‖δA‖ = 2‖A‖. By the inner product 〈Ax0,x0〉 = 0 ∈ Wess(A),it follows that ‖δA‖ = 2‖A‖. Therefore, ‖AY0 − Y0A‖ = 2‖A‖ = ‖δA‖. Hence, δA is norm-attainable. �
Lemma 20. Let A,A0 ∈ B(H). If there exists unit vectors y and y0 on H such that A,A0 are
norm-attainable then δA,A0 is also norm-attainable.

Proof. Given the operators A,A0 ∈ B(H) are norm-attainable then we need to show that δA,A0 isalso norm-attainable. We define the generalized derivation by δA,A0 = AY −Y A0. Since A,A0 arenorm-attainable then there exists unit vectors y and y0 on H such that ‖y‖ = ‖y0‖ = 1, ‖Ay‖ = ‖A‖and ‖A0y0‖ = ‖A0‖. By linear dependence of vectors, if y and Ay are linearly dependent then wehave ‖Ay‖ = η‖A‖y where |η| = 1 and |〈Ay,y〉| = ‖A‖. It follows that |〈A0y0,y0〉| = ‖A0‖ whichimplies that ‖A0y0‖ = φ‖A0‖y0 and |φ| = 1. Therefore, 〈A0y0‖A0‖,y0〉 = φ = −〈 Ay‖A‖,y〉 = −η. If Y is

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Eur. J. Math. Anal. 10.28924/ada/ma.3.9 11
defined as Y : y → y0 and y0 → 0, ‖Y‖ = 1 then (AY −Y A0)y0 = φ(‖A‖+‖A0‖)y0 which implies
‖AY −Y A0‖ = ‖(AY −Y A0)y0‖ = ‖A‖ + ‖A0‖ = ‖δA,A0‖. Hence, δA,A0 is norm-attainable. �
Lemma 21. Every inner derivation is norm-attainable if and only if it is self-adjoint.

Proof. Let δA ∈ B(H) be norm-attainable then we show that δA = δ∗A. Now since δA ∈ B(H)is norm-attainable then there exists a contraction Y ∈ B(H) such that ‖δAY‖ = ‖δA‖. That is,
‖δ∗AδAY‖ = ‖δ

2
AY‖. Let η ∈ H be defined as η = δA‖δA‖ then η is contractive such that ‖δ∗Aη‖ =

‖δA‖ = ‖δ∗A‖. Hence, δA is self-adjoint. Conversely, let δA be self-adjoint. Now since δ∗A is norm-attainable from the first part, then there exists a contractive M ∈ B(H) such that ‖δ∗AM‖ = ‖δ∗A‖, i.e
‖δAδ∗AM‖ = ‖δ

2
AM‖. Let ζ be denoted by ζ = δ∗A‖δ∗

A
‖ where ‖ζ‖ = 1 such that ‖δAζ‖ = ‖δ∗A‖ = ‖δA‖.Hence, δA is norm-attainable. �

Lemma 22. Every generalized derivation is norm-attainable if and only if it is implemented by
orthogonal projections.

Proof. Let A,A0 ∈ B(H) be orthogonal projections. Indeed, to show that a generalized derivationis implemented by orthogonal projections A and A0, it is enough to show that it is self-adjoint ifand only if it is normal as proved in [22]. Let δA,A0 : B(H) → B(H) be bounded linear operator on
B(H). Then exists a unique bounded linear operator δ∗A,A0 : B(H) → B(H) such that 〈δA,A0X,Y 〉 =
〈X,δ∗A,A0Y 〉, for all X,Y ∈ B(H). Now,

‖δ∗A,A0Y‖ = sup
‖X‖=1

〈δA,A0X,Y 〉

≤ sup
‖X‖=‖Y‖=1

‖δA,A0‖‖X‖‖Y‖

= ‖δA,A0‖

So, we conclude that δ∗A,A0 is norm-attainable. Conversely, let δA,A0 be norm-attainable. We needto show that it is implemented by orthogonal projections. This follows immediately from [22] andthis completes the proof. �
At this point, we give results on upper norm estimates for norm-attainable derivations. We con-sider both inner derivations and generalized derivations. We begin with the following proposition.

Proposition 23. Let A,B ∈ NA(H) and δA,B be bounded then ‖δA,B‖≤‖A‖ + ‖B‖.

Proof. Since δA,B is bounded then for fixed A,B ∈ NA(H) we have ‖δA,B(X)‖ ≤ ‖AX −XB‖ ≤
‖AX‖ + ‖XB‖ ≤ ‖A‖‖X‖ + ‖X‖‖B‖. Let X be of norm 1 and take supremum over X ∈ NA(H)then ‖δA,B‖≤‖A‖ + ‖B‖. �
Remark 24. If A = B then ‖δA‖≤ 2‖A‖.

Next, we consider upper bounds in the unit ball of NA(H) denoted by [NA(H)]0.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.9 12
Lemma 25. Let [NA(H)]0 be the unit ball of NA(H) and S be a fixed element of NA(H). Let
X ∈ [NA(H)]0 then ‖δS|[NA(H)]0‖≤ 2d(S).

Proof. Since X ∈ [NA(H)]0 has norm 1 then we have ‖δS|[NA(H)]0(X)‖ = ‖SX −XS‖[NA(H)]0 =
‖(S−λ)X−X(S−λ)‖[NA(H)]0 ≤‖S−λ‖‖X‖[NA(H)]0 +‖X‖‖S−λ‖[NA(H)]0. Taking the supremumover [NA(H)]0, we obtain ‖δS|[NA(H)]0‖ ≤ 2‖S −λ‖ and considering the infimum over λ ∈ C weobtain ‖δS|[NA(H)]0‖≤ 2 infλ∈C‖S −λ‖ = 2d(S). �
Remark 26. The restriction of δA|[NA(H)]0 i.e δA to [NA(H)]0 is a bounded linear operator.

Next we give an extension of Lemma 25 to a generalized derivation in the following theorem.
Theorem 27. Let S,S0 be fixed elements of NA(H) then ‖δS,S0|[NA(H)]0‖≤‖δS,S0‖.

Proof. Since X ∈ [NA(H)]0 has norm 1 then we have ‖δS,S0|[NA(H)]0(X)‖ = ‖SX−XS0‖. Followingproof of lemma 25 anologously we have
‖δS,S0|[NA(H)]0(X)‖≤‖S −λ‖‖X‖[NA(H)]0 + ‖X‖‖S0 −λ‖[NA(H)]0.

Taking the supremum over X ∈ [NA(H)]0 we obtain
‖δS,S0|[NA(H)]0‖≤ infλ∈C(‖S −λ‖ + ‖S0 −λ‖) = ‖δS,S0‖. �

Corollary 28. Every generalized derivation δS,S0 is norm-bounded.

Proof. This follows immediately from [49] and from Theorem 27. This completes the proof. �
Now, we consider lower bounds for norms of derivations. We begin the following proposition ongeneralized derivation.

Proposition 29. Let S,S0 be fixed elements of NA(H) then
‖δS,S0|[NA(H)]0‖≥‖S‖ + ‖S0‖.

Proof. Let η,ξ and x be unit vectors in H and φ,ϕ be positive linear functionals such that φ⊗η :
H → C and ϕ ⊗ ξ : H → C be of rank 1 defined as (φ ⊗ η)x = φ(x)η and (ϕ ⊗ ξ)x =
ϕ(x)ξ,∀x ∈ H,‖x‖ = 1. Now we have that ‖(φ⊗η)x‖ = sup{‖(φ⊗η)x‖,‖x‖ = 1} = |φ(x)| = |φ|.Similarly, we have ‖(ϕ ⊗ ξ)x‖ = ‖ϕ‖. Letting S = φ ⊗ η and S0 = ϕ ⊗ ξ then ‖S‖ = ‖φ‖and ‖S0‖ = ‖ϕ‖. Now from Corollary 28 we have that every generalized derivation is norm-bounded this implies that ‖δS,S0|[NA(H)]0(X)‖ ≥ ‖δS,S0(X)‖ where X ∈ [NA(H)]0. Therefore,
‖δS,S0|[NA(H)]0‖

2 ≥‖SX−XS0‖2 implying that ‖δS,S0|[NA(H)]0‖2 ≥ [‖S‖+‖S0‖]2. Taking positivesquare root on both sides we obtain ‖δS,S0|[NA(H)]0‖ = ‖δS,S0‖≥‖S‖ + ‖S0‖. �
Remark 30. If S = S0 then ‖δS,S0‖ = ‖δS‖≥ 2‖S‖.

Remark 31. From Theorem 27 and Proposition 3 it is easy to see that ‖δS,S0‖ = ‖S‖ + ‖S0‖ and
hence ‖δS‖ = 2‖S‖.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.9 13
Theorem 32. Let S,S0 ∈ NA(H) and α1 ∈ W0(S) and α2 ∈ W0(S0). Then ‖δS,S0‖ ≥ (‖S‖

2 −
|α1|2)1/2 + (‖S0‖2 −|α2|2)1/2.

Proof. By definition of W0(S) we have xn ∈ H such that ‖Sxn‖ = ‖S‖ and 〈Sxn,xn〉 → α1for α1 ∈ W0(S). This argument follows for W0(S0) and α2 ∈ W0(S0). Let Sxn = δnxn + βnynso S0xn = σnxn + λnyn where 〈xn,yn〉 = 0,‖yn‖ = 1. Take Unxn = xn and Unyn = −yn for
Un = 0 in {xn,yn}. Then ‖SUnxn − UnS0xn‖ = ‖δn + βn‖ ≤ |δn| + |βn|. But |δn| + |βn| ≥
(‖S‖2 −|δn|2)1/2 −ξn + (‖S0‖2 −|βn|2)1/2 −ξn). Since ξn is arbitrary and letting n →∞, so itfollows that ‖δS,S0‖≥‖(SUn−UnS0)xn‖ = |δn|+|βn| = (‖S‖2−|α1|2)1/2+(‖S0‖2−|α2|2)1/2. �
Corollary 33. Let 〈xn,yn〉 = 0 then 0 ∈ W0(S) and if 0 ∈ W0(S0) then ‖δS,S0‖≥‖S‖ + ‖S0‖.

Proof. Follows immediately from definition of W0(S) and the Theorem 32. �
4. Conclusion

In this paper, we have given a detailed characterization of operators in terms of norm-attainabilityconditions and norm estimates for in Banach algebras. In particular, we have established norm-attainability conditions for the derivations and also given the norm bounds in the norm-attainableclasses.
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https://doi.org/10.2140/pjm.1991.147.365
https://doi.org/10.1090/proc/12664
https://doi.org/10.1017/s0013091504000306

	1. Introduction
	2. Preliminaries
	3. Main Results
	4. Conclusion
	References

