Norm Attaining Research Articles

Norm attaining vectors and Hilbert points

Norm attaining vectors and Hilbert points

On the Pythagorean orthogonality of Hilbert bounded linear operators

In this work, we characterize the Pythagorean orthogonality of bounded linear operators using the concept of norm attainment set and disjoint support in Hilbert spaces.

Minimal boundaries for operator algebras

We study boundaries for unital operator algebras. These are sets of irreducible ∗ * -representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of C ∗ \mathrm {C}^* -algebras, our techniques allow us to provide a new proof of a recent characterization of those C ∗ \mathrm {C}^* -algebras admitting only finite-dimensional irreducible representations.

Several Remarks on Norm Attainment in Tensor Product Spaces

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if Xwidehat{otimes }_pi Y is strongly subdifferentiable and either X or Y has the metric approximation property then every bounded operator from X to Y^* is compact. We also prove that (ell _p(I)widehat{otimes }_pi ell _q(J))^* has the w^*-Kadec-Klee property for every non-empty sets I, J and every 2<p,q<infty , obtaining in particular that the norm of the space ell _p(I)widehat{otimes }_pi ell _q(J) is strongly subdifferentiable. This extends several results of Dantas, Kim, Lee and Mazzitelli. We also find examples of spaces X and Y for which the set of norm-attaining tensors in Xwidehat{otimes }_pi Y is dense but whose complement is dense too.

On a Set of Norm Attaining Operators and the Strong Birkhoff–James Orthogonality

Continuing the study of recent results on the Birkhoff–James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia–Šemrl property for operators which is weaker than the Bhatia–Šemrl property. The set of operators with the adjusted Bhatia–Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia–Šemrl property. It is known that the set of operators with the Bhatia–Šemrl property is norm-dense if the domain space X of the operators has the Radon–Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $$c_0$$ , $$L_1[0,1]$$ and C[0, 1]. In contrast with the Bhatia–Šemrl property, we show that the set of operators with the adjusted Bhatia–Šemrl property is norm-dense when the domain space is $$c_0$$ or $$L_1[0,1]$$ . Moreover, we show that the set of functionals having the adjusted Bhatia–Šemrl property on C[0, 1] is not norm-dense but such a set is weak- $$*$$ -dense in $$C(K)^*$$ for any compact Hausdorff K.

Norm attaining Lipschitz maps toward vectors

We characterize the norm attainment toward vectors for vector-valued Lipschitz maps defined on a general metric space. The main theorem of the present paper states that on a large class of metric spaces including infinite subsets of finite-dimensional spaces, every Lipschitz map attains its norm toward a vector if and only if the range space is finite-dimensional. Furthermore, motivated by the first negative example given by G. Godefroy, some denseness results for norm attaining Lipschitz maps toward vectors are also presented.

Some remarks on orthogonality of bounded linear operators-II

Let $$\mathbb{X,Y}$$ be normed linear spaces. We continue exploring the validity of the Bhatia–Šemrl (BŠ) Property: “An operator $$T \in \mathbb{L(X,Y)}$$ satisfies Bhatia–Šemrl Property if for any $$A \in \mathbb{L(X,Y)},T \bot_{B} A$$ implies that there exists a unit vector $$x \in \mathbb{X} $$ such that $$\lVert T_{x} \lVert = \lVert T \lVert $$ and $$T_{x}\bot_{B}A_{x}$$ .” A pair of normed linear spaces $$\mathbb{X,Y}$$ is said to be a BŠ pair if for every $$T\in \mathbb{L(X,Y),T}$$ satisfies the BŠ Property if and only if $$M_{T} = D{\cup}(-D)$$ , where D is a non-empty connected subset of $$S_{\mathbb{X}} $$ . We show that $$\ell^{n}_{1},\mathbb{Y}$$ is a BŠ pair for any normed linear space $$\mathbb{Y}$$ , and also obtain some other results in this context. In particular, using the notion of norm attainment set,we characterize the space $$\ell^{3}_{\infty}$$ among all 3-dimensional polyhedral Banach spaces whose unit ball have exactly eight extreme points.

On the closure of absolutely norm attaining operators

ABSTRACTLet H1 and H2 be complex Hilbert spaces and T:H1→H2 be a bounded linear operator. We say T is norm attaining if there exists x∈H1 with ‖x‖=1 such that ‖Tx‖=‖T‖. If for every non-zero closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. If we replace the norm of the operator by the minimum modulus m(T)=inf{‖Tx‖:x∈H1,‖x‖=1} in the above definitions, then T is called a minimum attaining and an absolutely minimum attaining operator or AM-operator, respectively. In this article, we discuss the operator norm closure of AN-operators. We completely characterize operators in this closure and study several important properties. We mainly give a spectral characterization of positive operators in this class and give a representation when the operator is normal. Later, we also study the analogous properties for AM-operators and prove that the closure of AM-operators is the same as the closure of AN-operators. Consequently, we prove similar results for operators in the norm closure of AM-operators.

Absolutely norm attaining Toeplitz and absolutely minimum attaining Hankel operators

Let H1 and H2 be complex Hilbert spaces. A bounded linear operator T:H1→H2 is called norm attaining if ‖Tx‖=‖T‖ for some unit vector x∈H1. If for every closed subspace M of H1, the restriction T|M:M→H2 is norm attaining, then T is called an absolutely norm attaining operator (or AN-operator). In the above definitions, if we replace the norm of the operator by the minimum modulus m(T)=inf⁡{‖Tx‖:x∈H1,‖x‖=1}, then T is called a minimum attaining and an absolutely minimum attaining operator (or AM-operator), respectively. In this article, we characterize Toeplitz AN-operators and discuss a few results on the minimum modulus of Toeplitz operator Tφ, φ∈L∞(T). We further characterize the minimum attaining Hankel operators and deduce that the only Hankel AM-operators are finite rank operators. While proving our results, we also obtained the following result; If φ∈L∞(T), then m(Lφ)=ess inf|φ| and there exists ψ∈L∞(T) such that γ(Lψ)>ess inf|ψ|, which improves a result from [15].

On quasi norm attaining operators between Banach spaces

We provide a characterization of the Radon–Nikodým property for a Banach space Y in terms of the denseness of bounded linear operators into Y which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970s for domain spaces. To this end, we introduce the following notion: an operator T:X longrightarrow Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence (x_n) of norm one elements in X such that (Tx_n) converges to some uin Y with Vert uVert =Vert TVert . We prove that strong Radon–Nikodým operators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators. It shows that this new notion of quasi norm attainment allows us to characterize the Radon–Nikodým property in terms of denseness of quasi norm attaining operators for both domain and range spaces, which in the case of norm attaining operators, was only valid for domain spaces due to the celebrated counterexample by Gowers in 1990. A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining nonlinear maps, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions such that quasi norm attaining operators are actually norm attaining, study the relation with the norm attainment of the adjoint operator and, finally, present some stability results.

Norm attainment set and symmetricity of operators on $$\ell _p^2$$

Norm attainment set and symmetricity of operators on $$\ell _p^2$$

A representation of hyponormal absolutely norm attaining operators

In this article, we characterize absolutely norm attaining normal operators in terms of the essential spectrum. Later we prove a structure theorem for hyponormal absolutely norm attaining (or AN-operators in short) and deduce conditions for the normality of the operator.

Norm derivatives and geometry of bilinear operators

We study the norm derivatives in the context of Birkhoff–James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff–James orthogonality in \(\ell _1^n\) and \(\ell _\infty ^n\) spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff–James orthogonality. We further study Birkhoff–James orthogonality, approximate Birkhoff–James orthogonality, smoothness and norm attainment of bounded bilinear operators between Banach spaces.

The norm attainment problem for functions of projections

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. Every skew projection on that Hilbert space is contained in such an algebra and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.

On a subclass of norm attaining operators

A bounded linear operator T: H1 → H2, where H1, H2 are Hilbert spaces, is said to be norm attaining if there exists a unit vector x ∈ H1 such that kT xk = kT k and absolutely norm attaining (or AN -operator) if T |M: M → H2 is norm attaining for every closed subspace M of H1. We prove a structure theorem for positive operators in β(H):= {T ∈ B(H): T |M: M → M is norm attaining for all M ∈ RT }, where RT is the set of all reducing subspaces of T . We also compare our results with those of absolutely norm attaining operators. Later, we characterize all operators in this new class.

Some Results on the Norm Attainment Set for Bounded Linear Operators on Smooth Banach Spaces

In this paper, we give new results and proofs that include the notion of norm attainment set of bounded linear operators on a smooth Banach spaces and using these results to characterize a bounded linear operators on smooth Banach spaces that preserve of approximate - -orthogonality. Noting that this work takes brief sidetrack in terms of approximate - -orthogonality relations characterizations of a smooth Banach spaces.

The Birkhoff-James orthogonality and norm attainment for multilinear maps

Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-Šemrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show that the denseness of operators does not hold when the domain space is c0 for arbitrary range. Moreover, we find plenty of Banach spaces Y such that only the zero operator has the Bhatia-Šemrl property in the space of operators from c0 to Y.

Some remarks on the generalization of orthogonality in terms of operators

This paper deals with a generalization of orthogonality in terms of bounded linear operators on a Banach space. The goal is to find a relation between orthogonality of images and orthogonality of elements. We prove that if the images of a bounded linear operator are orthogonal in the Pythagorean sense, then the elements are orthogonal in the sense of Birkhoff's definition. In the case of Robert's orthogonality in terms of bounded linear operators under the restriction that any element belongs to the intersection of the norm attainment set of $T_1+\lambda T_2$ and $T_1-\lambda T_2$, if the images are orthogonal, then it implies that the operators are also orthogonal. Furthermore, some results in relation to the Carlsson, isosceles, and approximate Birkhoff-James orthogonality have been obtained. <br><br> У цій роботі розглядяється узагальнення ортогональності в термінах обмежених лінійних операторів на банаховому просторі. Метою роботи є знайти співвідношення між ортогональністю зображень і ортогональністю елементів. Доведено, що якщо образи обмеженого лінійного оператора ортогональні в піфагоровому сенсі, то елементи є ортогональний у сенсі визначення Біркгофа. У випадку ортогональності Роберта в термінах обмежених лінійних операторів із умовою, що будь-який елемент належить до перетину множин де оператори $T_1+\lambda T_2$ і $T_1-\lambda T_2$ досягають норми, з ортогональності образів випливає, що оператори є також ортогональні. Також отримано деякі результати про ортогональність в сенсі Карлссона, рівнобічної ортогональності та наближеної ортогональності в сенсі Біркгофа-Джеймса.

Characterization of Absolutely Norm Attaining Compact Hyponormal Operators

In this paper, we characterize absolute norm-attainability for compact hyponormal operators. We give necessary and sufficient conditions for a bounded linear compact hyponormal operator on an infinite dimensional complex Hilbert space to be absolutely norm attaining. Moreover, we discuss the structure of compact hyponormal operators when they are self-adjoint and normal. Lastly, we discuss in general, other properties of compact hyponormal operators when they are absolutely norm attaining and their commutators.

Orthogonality and norm attainment of operators in semi-Hilbertian spaces

We study the semi-Hilbertian structure induced by a positive operator A on a Hilbert space \({\mathbb {H}}.\) Restricting our attention to \(A-\)bounded positive operators, we characterize the norm attainment set and also investigate the corresponding compactness property. We obtain a complete characterization of the \(A-\)Birkhoff–James orthogonality of \(A-\)bounded operators under an additional boundedness condition. This extends the finite-dimensional Bhatia-\( \breve{S} \)emrl Theorem verbatim to the infinite-dimensional setting.