Search

In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space H . Among other results, we show that if H is a finite-dimensional Hilbert space and T ∈ B (H) , then T is self-adjoint if and only if there exists p > 0 such that | T | p ≤ | Re (T) | p . If in addition, T and Re T are invertible, then T is self-adjoint if and only if log | T | ≤ log | Re (T) | . Considering the polar decomposition T = U | T | of T ∈ B (H) , we show that T is self-adjoint if and only if T is p-hyponormal (log-hyponormal) and U is self-adjoint. Also, if T = U | T | ∈ B (H) is a log-hyponormal operator and the spectrum of U is contained within the set of vertices of a regular polygon, then T is necessarily normal. [ABSTRACT FROM AUTHOR]

Let A be a positive (semidefinite) bounded linear operator on a complex Hilbert space (H , ⟨ · , · ⟩) . The semi-inner product induced by A is defined by ⟨ x , y ⟩ A : = ⟨ A x , y ⟩ for all x , y ∈ H and defines a seminorm ‖ · ‖ A on H . This makes H into a semi-Hilbert space. For p ∈ [ 1 , + ∞) , the generalized A-joint numerical radius of a d-tuple of operators T = (T 1 , ... , T d) is given by ω A , p (T) = sup ‖ x ‖ A = 1 ∑ k = 1 d | 〈 T k x , x 〉 A | p 1 p. Our aim in this paper is to establish several bounds involving ω A , p (·) . In particular, under suitable conditions on the operators tuple T , we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005). [ABSTRACT FROM AUTHOR]

In this paper, it is investigated that necessary and sufficient conditions for a minimal operator defined by a degenerate first-order differential operator expression in the Hilbert space to be formally normal. Also, all normal extensions of the minimal operator are given with their domains. Moreover, the spectrum set of these normal extensions is given through the family of evolution operators. [ABSTRACT FROM AUTHOR]

The concept of adjusted sublevel set for a quasiconvex function was introduced by Aussel and Hadjisavvas who proved the local existence of a norm-to-weak ∗ upper semicontinuous base-valued submap of the normal operator associated with the adjusted sublevel set. When the space is finite-dimensional, a globally defined upper semicontinuous base-valued submap is obtained by taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique does not work in the infinite-dimensional case. We propose a partition of unity technique to overcome this problem in Banach spaces. An application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

In this work, we give several upper bounds for the numerical radius of the sum of two operators as well as for the numerical radii of 2 × 2 operator matrices. These upper bounds refine some earlier existing ones. [ABSTRACT FROM AUTHOR]

In this research article the necessary and sufficient conditions for the norm of composition operator C Φ on A α 2 (H) to be one are obtained. Moreover, C Φ is unitary on A α 2 (H) if and only if it is co-isometry. The necessary and sufficient condition for Hermitian and normal composition operators on A α 2 (H) are also explored. Also, the characterization for boundedness of generalized weighted composition operator is obtained under some condition on Φ. [ABSTRACT FROM AUTHOR]

In this paper, we mainly show that if a product AB (or BA) of a closed symmetric operator A and a bounded positive operator B is normal, then it is self-adjoint. Equivalently, this means that B commutes with A. Certain generalizations and consequences are also presented. [ABSTRACT FROM AUTHOR]

This work deals with some generalizations of the Drazin invertibility, involving the pseudo-generalized invertibility that has been studied recently by the authors in Lahmar and Skhiri (Filomat 36:2551-2572, 2022) and Lahmar and Skhiri (Filomat 36:4575-4590, 2022). We also introduce a new notion of DPG inverse and the relationships between the notions of DPG inverse, Drazin inverse and EP operator, are studied in this paper. An application of DPG inverses to some operator equations is also provided. [ABSTRACT FROM AUTHOR]

A bounded linear operator T on a complex Hilbert space ℋ is called n-normal if T*Tn = TnT*. By Fuglede's theorem T is n-normal if and only if Tn is normal. Let k, n ∈ ℕ. Then a bounded linear operator T is said to be of type I k-quasi-n-normal if T*k{T*Tn − TnT*}Tk =0, and T is said to be of type II k-quasi-n-normal if T*k{T*nTn − TnT*n}Tk =0. 1-quasi-n-normal is called quasi-n-normal. We shall show that (1) type I quasi-2-normal and type II quasi-2-normal are different classes; (2) the intersection of the class of type I quasi-2-normal and the class of type II quasi-2-normal is equal to the class of 2-normal. We also give some examples of type I k-quasi-n-normal and type II k-quasi-n-normal. We also show that Weyl's theorem holds for this class of operators and every k-quasi-n-normal operator has a non trivial invariant subspace. [ABSTRACT FROM AUTHOR]

We establish all normal extensions domains structure on the weighed Hilbert space L α 2 (H , (0 , 1)) for a formally normal minimal operators class defined by a degenerate conformable α -order differential-operator (B u) (α) (t) + A u (t) , α , t ∈ (0 , 1) , where a positive selfadjoint B : H → H has a closed range, KerB is a nontrivial subspace and A : H → H is a selfadjoint operator. In addition, it is given the analysis of the spectrum set for any normal extensions. [ABSTRACT FROM AUTHOR]

Hankel operators have numerous realizations and form one of the most important classes of operators in the spaces of analytic functions. These operators can be defined as those having Hankel matrices (i.e. matrices whose entries depend only on the sum of the indices) with respect to some orthonormal basis for a separable Hilbert space. This paper continues the authors' research of 2021 which introduced a new class of operators in Hilbert spaces; i.e., -Hankel operators, with a complex parameter. These operators act in a separable Hilbert space and have matrices in some orthonormal basis of the space whose diagonals orthogonal to the main diagonal present geometric progressions with common ratio . Thus, the classical Hankel operators correspond to the case . The main result of the article is the normality criterion for -Hankel operators. By analogy with Hankel operators, the class of operators under consideration has concrete realizations in the form of integral operators which enables us to apply the abstract results, and thereby contribute to the theory of integral operators. We consider an example of these realizations in the Hardy space on the unit disk. Also, we give some criteria for the self-adjointness and normality of these operators. [ABSTRACT FROM AUTHOR]

We study the unitary orbit of a normal operator a ∈ B (H) , regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the various ambient spaces if and only if the spectrum of a is finite, and in that case, it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of a is not finite, we describe the closure of the orbits of a for the different norm topologies involved. We relate these results to the action of the groupoid of partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator a, we also describe the norm closure of its groupoid orbit O a , which leads to necessary and sufficient spectral conditions on a ensuring that O a is norm closed and that O a is a closed embedded submanifold of B (H) . [ABSTRACT FROM AUTHOR]

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems. [ABSTRACT FROM AUTHOR]

We characterize the normal operators A on ℓ 2 and the elements a i ∈ ℓ 2 , with 1 ≤ i ≤ m , such that the sequence { A n a 1 , … , A n a m } n ≥ 0 is a frame. The characterization makes strong use of the pseudo-hyperbolic metric of D and is given in terms of the backward shift invariant subspaces of H 2 (D) associated to finite products of interpolating Blaschke products. [ABSTRACT FROM AUTHOR]

In this paper we aim to study the tensor product and the tensor sum of two jointly-normal operators. Mainly, an alternative proof is given for the result of Chō and Takaguchi (Pac J Math 95(1):27–35, 1981) asserting that: if T is jointly-normal, then r (T) = ‖ T ‖ = ω (T) , where r (T) , ω (T) and ‖ T ‖ denote respectively the joint spectral radius, the joint numerical radius and the joint norm of an operator tuple T . It seems that this new method allows to handle more general situations, namely the operators acting on semi-hilbertian spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a polynomially normal operator on a complex Hilbert space, extending the notation of n-normal and normal operators. Several basic properties of polynomially normal operator are firstly presented. We show some spectral properties of polynomially normal operators under new assumption in literature. Precisely, we prove that σ (T) = σ a (T) , ker (T - z) ⊥ ker (T - w) if z and w are distinct eigen-values of T and others results. Thus, we generalize some results for n-normal and normal operators. [ABSTRACT FROM AUTHOR]

In this paper, we discuss measure theoretic characterizations for Lambert conditional operators in some operator classes on L 2 (Σ) such as, p-hyponormal, centered, n-normal and binormal. In addition, it is showed when these operators are orthogonal projection and some correlations between these types of operators are established. [ABSTRACT FROM AUTHOR]

In this paper, we focus on properties of the dilation of truncated Toeplitz operators on L 2 . In particular, we provide necessary and sufficient conditions for the dilation of truncated Toeplitz operators to be nonnegative and self-adjoint. Finally, we study the symbols φ and ψ when the dilation of truncated Toeplitz operators, S φ , ψ (defined below), is normal. [ABSTRACT FROM AUTHOR]

We consider generalized Hausdorff operators and introduce the notion of the symbol of such an operator. Using this notion, we describe, under some natural conditions, the structure and investigate important properties (such as invertibility, spectrum, and norm) of normal generalized Hausdorff operators on Lebesgue spaces over ℝn. As an example we consider Cesàro operators. [ABSTRACT FROM AUTHOR]

In this paper we study Grüss type inequalities for real and complex valued functions in probability spaces. Some earlier Grüss type inequalities are extended and refined. Our approach leads to new integral inequalities which are interesting in their own right. As an application, we give a Grüss type inequality for normal operators in a Hilbert space. Similar results are obtained only for self-adjoint operators in earlier papers. [ABSTRACT FROM AUTHOR]

Quasiconvex functions present some difficulties in global optimization, because their graph contains “flat parts”; thus, a local minimum is not necessarily the global minimum. In this paper, we show that any lower semicontinuous quasiconvex function may be written as a composition of two functions, one of which is nondecreasing, and the other is quasiconvex with the property that every local minimum is global minimum. Thus, finding the global minimum of any lower semicontinuous quasiconvex function is equivalent to finding the minimum of a quasiconvex function, which has no local minima other than its global minimum. The construction of the decomposition is based on the notion of “adjusted sublevel set.” In particular, we study the structure of the class of sublevel sets, and the continuity properties of the sublevel set operator and its corresponding normal operator. [ABSTRACT FROM AUTHOR]

The characterization of normal truncated Toepltiz operators is first given by Chalendar and Timotin. We give an elementary proof of their result without using the algebraic properties of truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

In this note, we consider the question of when a Toeplitz operator on the Hardy-Hilbert space $$H^2$$ of the open unit disk $$\mathbb {D}$$ is complex symmetric, focusing on symbols $$\phi :\mathbb {T}\rightarrow \mathbb {C}$$ that are continuous on the unit circle $$\mathbb {T}=\partial \mathbb {D}$$ . A closed curve $$\phi $$ is called nowhere winding if the winding number of $$\phi $$ is 0 about every point not in the range of $$\phi $$ . It is then shown that if $$T_\phi $$ is complex symmetric, then $$\phi $$ must be nowhere winding. Hence if $$\phi $$ is a simple closed curve, then $$T_\phi $$ cannot be a complex symmetric operator. The spectrum and invertibility of complex symmetric Toeplitz operators with continuous symbols are then described. Finally, given any continuous curve $$\gamma :[a,b]\rightarrow \mathbb {C}$$ , it is shown that there exists a complex symmetric Toeplitz operator with continuous symbol whose spectrum is precisely the range of $$\gamma $$ . [ABSTRACT FROM AUTHOR]

Algorithms of the Bartels-Stewart type for the numerical solution of Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are normal. The superiority of the modified algorithms over the original ones is illustrated by numerical results. [ABSTRACT FROM AUTHOR]

An operator $$S_{\varphi ,\psi }^{u}\in \mathcal {L}(L^2)$$ is called the dilation of a truncated Toeplitz operator if for two symbols $$\varphi ,\psi \in L^{\infty }$$ and an inner function u, holds for $$f\in {L}^{2}$$ where $$P_{u}$$ denotes the orthogonal projection of $$L^2$$ onto the model space $$\mathcal { K}_{u}^2=H^2{\ominus }{{u}H^2}$$ and $$Q_u=I-P_u.$$ In this paper, we study properties of the dilation of truncated Toeplitz operators on $$L^{2}$$ . In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators. [ABSTRACT FROM AUTHOR]

Positivity of a 2 × 2 operator matrix × [ABBC] ≥ 0 implies ‖A‖ . ‖C‖ ≥ ‖B‖ for operator norm ‖ . ‖. This can be considered as an operator version of the Schwarz inequality. In this situation, for A;C ≥ 0, there is a natural notion of geometric mean AC, for which √ ‖A‖ . ‖C‖ . ‖AC‖. In this paper, we study under what conditions on A, B, and C or on B alone the norm inequality √‖A‖ . ‖C‖ . ‖B‖ can be improved as ‖AC‖ ≥ ‖B‖. [ABSTRACT FROM AUTHOR]

The operator defined by the left-hand side of the matrix equation is not linear in the space of complex matrices $$X$$ . However, it becomes a linear operator if every matrix $$X$$ is replaced by the pair of real matrices of the same size, namely, by its real and imaginary parts. Introducing an appropriate scalar product in the resulting real space, we find necessary and sufficient conditions for this operator to be normal. [ABSTRACT FROM AUTHOR]

In this paper, we study the Fuglede-Putnam property. We give a necessary and sufficient condition for which $$(\oplus _{i=1}^{n}A_i,\oplus _{i=1}^{n}B_i)$$ satisfies the Fuglede-Putnam property. We also study the local spectral theory associated with the Fuglede-Putnam property. Finally, we define the weak Fuglede-Putnam property and we investigate several cases which satisfy the weak Fuglede-Putnam property. [ABSTRACT FROM AUTHOR]

Inspired by similar definition in subdifferential theory, we define limiting sublevel set and limiting normal operator maps for quasiconvex functions. These maps satisfy important properties as semicontinuity and quasimonotonicity. Moreover, calculus rules together with necessary and sufficient optimality conditions for constrained optimization are established. [ABSTRACT FROM AUTHOR]

Fundamental properties of unbounded composition operators in $$L^2$$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition operators generating Stieltjes moment sequences are completely characterized. The unbounded counterparts of the celebrated Lambert's characterizations of subnormality of bounded composition operators are shown to be false. Various illustrative examples are supplied. [ABSTRACT FROM AUTHOR]

Let α and β be functions in $${L^\infty(\mathbb{T})}$$ , where $${\mathbb{T}}$$ is the unit circle. Let P denote the orthogonal projection from $${L^2(\mathbb{T})}$$ onto the Hardy space $${H^2(\mathbb{T})}$$ , and Q = I − P, where I is the identity operator on $${L^2(\mathbb{T})}$$ . This paper is concerned with the singular integral operators S on $${L^2(\mathbb{T})}$$ of the form S f = αPf + βQf, for $${f \in L^2(\mathbb{T})}$$ . In this paper, we study the normality of S which is related to the Brown-Halmos theorem for the normal Toeplitz operator on $${H^2(\mathbb{T})}$$ . [ABSTRACT FROM AUTHOR]

Given a complex Krein space $${\mathcal{H}}$$ with fundamental symmetry J, the aim of this note is to characterize the set of J-normal projections The ranges of the projections in $${\mathcal{Q}}$$ are exactly those subspaces of $${\mathcal{H}}$$ which are pseudo-regular. For a fixed pseudo-regular subspace $${\mathcal{S}}$$, there are infinitely many J-normal projections onto it, unless $${\mathcal{S}}$$ is regular. Therefore, most of the material herein is devoted to parametrizing the set of J-normal projections onto a fixed pseudo-regular subspace $${\mathcal{S}}$$. [ABSTRACT FROM AUTHOR]

The concept of maximal quasimonotonicity of set-valued map is introduced and studied. Regularity properties of this class of operators is investigated, in particular the ccvc property, an adaptation to the quasimonotone case of the classical notion of cusco map. In an Asplund space, we provide sufficient conditions for a ccvc quasimonotone operator to be single-directional on a $$G_\delta $$ -dense subset. [ABSTRACT FROM AUTHOR]

A formally normal weighted shift on a directed tree is shown to be a bounded normal operator. The question of whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some, possibly different, directed tree is answered. [ABSTRACT FROM AUTHOR]

Let A = U| A| and B = V| B| be the polar decompositions of $${A \in \mathbb{B}(\fancyscript{H}_1)}$$ and $${B\in \mathbb{B}(\fancyscript{H}_2)}$$ and let Com( A, B) stand for the set of operators $${X \in \mathbb{B}(\fancyscript{H}_2,\fancyscript{H}_1)}$$ such that AX = XB. A pair ( A, B) is said to have the FP-property if Com( A, B) $${\subseteq}$$ Com( A*, B*). Let $${\tilde{C}}$$ denote the Aluthge transform of a bounded linear operator C. We show that (i) if A and B are invertible and ( A, B) has the FP-property, then so is $${(\tilde{A}, \tilde{B})}$$ ; (ii) if A and B are invertible, the spectrums of both U and V are contained in some open semicircle and $${(\tilde{A}, \tilde{B})}$$ has the FP-property, then so is ( A, B); (iii) if ( A, B) has the FP-property, then Com( A, B) $${\subseteq}$$ Com $${(\tilde{A},\tilde{B})}$$, moreover, if A is invertible, then Com( A, B) = Com $${(\tilde{A}, \tilde{B})}$$. Finally, if $${\mbox{Re}(U|A|^{1\over2})\geq a>0}$$ and $${\mbox{Re}(V|B|^{1\over2})\geq a>0}$$ and X is an operator such that U* X = XV, then we prove that $${\|\tilde{A}^* X-X\tilde{B}\|_p\geq 2a\|\,|B|^{1\over2}X-X|B|^{1\over2} \|_p}$$ for any $${1 \leq p \leq \infty}$$. [ABSTRACT FROM AUTHOR]

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are $${(1 - \overline{w}z)^{-\kappa}}$$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces. [ABSTRACT FROM AUTHOR]

Equations such as AB = B A have been studied in the finite dimensional setting in (Linear Algebra Appl 369:279-294, ). These equations have implications for the spectrum of B, when A is normal. Our aim is to generalize these results to an infinite dimensional setting. In this case it is natural to use JB* J for some conjugation operator J in place of B. Our main result is a spectral pairing theorem for a bounded normal operator B which is applied to the study of the equation KB = B* K for K an antiunitary operator. In particular, using conjugation operators, we generalize the notion of Hamiltonian operator and skew-Hamiltonian operator in a natural way, derive some of their properties, and give a characterization of certain operators B for which AB = ( JB* J) A and BA = A( JB* J) and also those B with KB = B* K for certain antiunitary operators K. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study generalized multiplication operators on weighted hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator $$\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}}$$ is complex symmetric, (ii) if T is a complex symmetric operator, then $$(\tilde{T})^{*}$$ and $$\widetilde{T^{*}}$$ are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then $$\widetilde{(T^{*})}_{s,t}$$ and $$(\tilde{T}_{t,s})^{*}$$ are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA( t, t), then T is normal. [ABSTRACT FROM AUTHOR]

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). [ABSTRACT FROM AUTHOR]

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with $$a_{1} \leq A \leq a_{2}$$ and $$b_{1} \leq B \leq b_{2}$$ for some real numbers a 1, a 2, b 1, and b 2, then for every unitarily invariant norm|||·|||, . If, in addition, X is positive, then . These norm inequalities generalize recent related inequalities due to Kittaneh, Bhatia-Kittaneh, and Wang-Du. [ABSTRACT FROM AUTHOR]

The article presents the lemmas, proof of the theorem, and corollaries for projection-convex combinations in C-algebras and the invariant subspace problem.

Let B(H) denote the algebra of operators on a complex separable Hilbert space H, and let A $\in$ B(H) have the polar decomposition A = U|A|. The Aluthge transform $$ \widetilde{A}\,\,\textrm{of} \,A $$ is defined to be the operator $$ \widetilde{A}=|A|^{\frac{1}{2}}U|A|^{\frac{1}{2}} $$. We say that A $\in$ B(H) is p-hyponormal, $$ 0 < p, \,\textrm{if}\,|A^{\ast}|^{2p}\leq |A|^{2p} $$. Let $$ \check{B} = \tilde{B}\ast^{\ast} $$. Given p-hyponormal $$ A, B^{\ast}, \frac{1}{2}\leq p < 1 $$ , such that AB is compact, this note considers the relationship between $$ |\lambda_{j}(AB)|, s_{j}(AB), |\lambda_{j}(\tilde{A}\check{B})|,s_{j}(\tilde{A}\check{B})\,\textrm{and}\,s_{j}(\check{B}\tilde{A}),\, \textrm{where}\,|\lambda_{j}(T)|\,(\textrm{respectively}, s_{j}(T)) $$ denotes an enumeration in decreasing order repeated according to multiplicity of the eigenvalues of the compact operator T (respectively, singular values of the compact operator T). It is proved that $$ s_{j}(AB) $$ is bounded above by $$ s_{j}(\tilde{A}\check{B}) $$ and below by $$ s_{j}(\check{B}\tilde{A}) $$ for all j = 1, 2, . . . and that if also $$ \tilde{A}\check{B} $$ is normal, then there exists a unitary U1 such that $$ |\lambda_{j}(U_{1}AB)| = |\lambda_{j}(\tilde{A}\breve{B})| = |\lambda_{j}(\breve{B}\tilde{A})| = s_{j}(\tilde{A}\check{B}) = s_{j}(\check{B}\tilde{A}) = s_{j}(AB) = s_{j}(U_{1}AB) $$ for all j = 1, 2, . . .. [ABSTRACT FROM AUTHOR]

In this short note, we give an elementary (set-theoretic) proof of Fuglede's theorem that the commutant of a normal operator is *-closed. [ABSTRACT FROM AUTHOR]

We investigate the commutation between a continuous linear random operator and a continuous linear deterministic operator on a Banach space. From this we obtain probabilistic versions of theorems by Fuglede and Putnam, both of them dealing with the commutation between continuous linear operators with continuous normal operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (self-adjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities. [ABSTRACT FROM AUTHOR]