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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]

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 paper, we study regular and Fredholm solvability of a Neumann type boundary value problem for a second order elliptic type equation with operator coefficients for a separable Hilbert space on a finite domain. The conditions of regular and Fredholm solvability for the given problem in terms of only the coefficients of the equation are found. The estimates for intermediate derivatives operators are obtained. These estimates determine the solvability conditions for our problem. Note that the considered operator equations have variable coefficients. [ABSTRACT FROM AUTHOR]

We examine Fuglede commutation properties, particularly those in the context of Lorentz ideals, from a new perspective. We also show that Fuglede commutation property fails for a number of ideals. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to determine when a partially isometric matrix is normal. However, we do not restrict ourselves to the finite-dimensional case and we describe when a partial isometry in ℬ ⁢ (ℋ) satisfies several strong and weak normal properties. In particular, we give elegant characterisations of normal partial isometries on infinite-dimensional Hilbert Spaces in terms of generalized inverses; we collect some spectral properties and explore when an operator in ℬ ⁢ (ℋ) is similar to a normal partial isometry. We close the paper by treating when a partial isometry is hyponormal. [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 paper, we study the structure of shift preserving operators acting on shift-invariant spaces in L²(G), where G is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from L²(ℝd) to L²(G). We investigate the matrix structure of range operator R(ξ) on range function J associated to shift-invariant space V, in the case of a locally compact Abelian group G. We also focus on some properties like as normal and unitary operator for range operator on L²(G). We show that shift preserving operator U is invertible if and only if fiber of corresponding range operator R is invertible and investigate the measurability of inverse R-1(ξ) of range operator on L²(G). [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]

The main focus of this paper is on the relationship between the spectrum of generators and the regularity of the fractional resolvent family. We will give a counter-example to show that the point-spectral mapping theorem is not valid for {Sα(t)} if α ≠ 1; and we show that if {Sα(t)} is stable, then we can determine the decay rate by σ(A) and some examples are given; we also prove that Sα(t)x has a continuous derivative of order αβ > 0 if and only if x ∈ D(I -- A)β. The main method we used here is the resolution of identity corresponding to a normal operator A and spectral measure integral. [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]

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.

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 work, we give refinements of some well-known numerical radius inequalities. Also, we present an improvement of the triangle inequality for the operator norm. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a right linear bounded operator on a quaternionic Hilbert space, the norm and the numerical radius are equal if and only if the norm and the spectral radius are equal. We also show that the spherical spectrum of a quaternionic bounded operator is included in the closure of its numerical range, and we show that the numerical range of an operator on a quaternionic Hilbert space is not necessarily convex. For a quaternionic bounded normal operator, we prove that the convex hull of the closure of its numerical range is equal to the convex hull of its spherical spectrum. Finally, we give some inequalities between the numerical radius, the spectral radius and the norm of a right linear bounded operator, and we prove also that the norm and the numerical radius of a quaternionic bounded hyponormal operator are equal. [ABSTRACT FROM AUTHOR]

In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These characterizations are useful in checking C-normality of an operator. We illustrate these results with examples. [ABSTRACT FROM AUTHOR]

The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we present new properties of polynomially normal operators. Precisely, we prove that under certain conditions polynomially normal operators are Drazin or even group invertible and we also give necessary and sufficient conditions for a polynomially normal operator to have a closed range. In addition, we characterize polynomially normal operators with or without closed ranges applying the adequate operator matrix representations. Furthermore, we show that in some cases a polynomially normal operator can be written as a direct sum of a normal operator and a nilpotent operator. What is more, we state several examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Let B (H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H . For A , B ∈ B (H) , define the bimultiplication operator M 2 , A , B on the class of Hilbert–Schmidt operators by M 2 , A , B (X) = A X B. In this paper, we show that if B is normal, then c o (W 0 (A) W 0 (B)) ⊆ W 0 (M 2 , A , B) , where co stands for the convex hull and W 0 (.) denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation δ 2 , A , B on the class of Hilbert–Schmidt operators are also given. [ABSTRACT FROM AUTHOR]

Let B (H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H . For A , B ∈ B (H) , define the bimultiplication operator M 2 , A , B on the class of Hilbert–Schmidt operators by M 2 , A , B (X) = A X B. In this paper, we show that if B is normal, then c o (W 0 (A) W 0 (B)) ⊆ W 0 (M 2 , A , B) , where co stands for the convex hull and W 0 (.) denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation δ 2 , A , B on the class of Hilbert–Schmidt operators are also given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of operators on separable Hilbert space called poly. normal operator. An operator is called poly. normal if there exists a polynomial ℙ, where such that , where * is the adjoint operator of. We investigated some basic properties and new results were discussed in this research. [ABSTRACT FROM AUTHOR]

Let (M , τ) be a semi-finite von Neumann algebra, L 0 (M) be the set of all τ-measurable operators, μ t (x) be the generalized singular number of x ∈ L 0 (M) and f : [ 0 , ∞) → [ 0 , ∞) be a concave function. We proved that if x 1 , x 2 , ... , x n are normal operators in L 0 (M) , then μ (f (| ∑ k = 1 n x k |)) is submajorized by μ (f (∑ k = 1 n | x k |)). As an application, we obtained that if x is a matrix of normal operators x i j in L 0 (M) , then μ (f (| x |)) is submajorized by μ (∑ i , j = 1 n f (| x i j |)). [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]

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the weighted Bergman space A α 2 (D). We first give properties of complex symmetric Toeplitz operators T φ on A α 2 (D). Next, we prove that if T φ is complex symmetric with finite symbol, then T φ is hyponormal on A α 2 (D) if and only if it is hyponormal on the Hardy space H 2 (T). Finally, we consider the complex symmetric Toeplitz operator T φ on A α 2 (D) when the conjugation is a special case. [ABSTRACT FROM AUTHOR]

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In this paper, unitary symmetrizers are considered. It is well known that using Newton operatoralgorithm, similar to the usual Newton algorithm, for extracting the square root, one can provethat for every Hermitian operator T 0, there exists a unique Hermitian operator S 0 suchthat T = S2. Moreover, S commutes with every bounded operator R with which commutes T. Theoperator S is called a square root of the operator T and is denoted by T1=2. The existence of thesquare root allows one to determine the absolute value jTj = (TT)1=2 of the bounded operator T.For every bounded linear operator T : H ! H there exists a unique partially isometric operatorU : H ! H such that T = UjTj, KerU = KerT. Such an equality is called a polar expansionof the operator T. The Schmidt operator is understood as the unitary multiplier of the polarexpansion of a compact inverse operator, with the help of which E. Schmidt was the rst to obtainthe expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers.This paper shows that the unitary symmetrizer of an operator diers only in sign from the adjointSchmidt operator. The main result of the paper: if A is an invertible and compact operator, andS is a unitary operator such that the operator SA is self-adjoint, then the operator AS is alsoself-adjoint and the formula S = U holds, where U is the Schmidt operator.

Let B(H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H. For A;B 2 B(H), define the bimultiplication operator M2;A;B on the class of Hilbert-Schmidt operators by M2;A;B(X) = AXB. In this paper, we show that if B, the adjoint operator of B, is hyponormal, then co(W0(A)W0(B)) W0(M2;A;B); where co stands for the convex hull and W0(:) denotes the maximal numerical range. If in addition, A is hyponormal, we show that co(W0(A)W0(B)) = W0(M2;A;B): [ABSTRACT FROM AUTHOR]

Let B(H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H. For A;B 2 B(H), define the bimultiplication operator M2;A;B on the class of Hilbert-Schmidt operators by M2;A;B(X) = AXB. In this paper, we show that if B, the adjoint operator of B, is hyponormal, then co(W0(A)W0(B)) W0(M2;A;B); where co stands for the convex hull and W0(:) denotes the maximal numerical range. If in addition, A is hyponormal, we show that co(W0(A)W0(B)) = W0(M2;A;B): [ABSTRACT FROM AUTHOR]

Let T be a normal bounded operator on a Hilbert space and let ω(T) denote the numerical radius of T. In this paper, we give new inequalities numerical radius of normal operatos on a Hilbert space, one of these inequalties we prove that where T = T1 + iT2 the Cartesian decomposition of T and. Moreover, some other related results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we give new results for the p-numerical radii w p ⋅ of Hilbert space operators. It is shown, among other inequalities, that if A is a Hilbert space operator, which belongs to the Schatten p-class, then w p p (A) ≥ w p / 2 p / 2 (A 2) 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 2 p + inf θ ∈ R Re ⁡ (e i θ A) p p − Im ⁡ (e i θ A) p p 2 and w p p (A) ≤ 2 p / 2 − 2 inf θ ∈ R Re ⁡ ((e i θ A) 2) p / 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 4 for 4 ≤ p < ∞. Also, w p p (A) ≥ w p / 2 p / 2 A 2 4 + A ∗ A + A A ∗ p / 2 p / 2 2 p / 2 + 2 + inf θ ∈ R Re ⁡ (e i θ A) p p − Im ⁡ (e i θ A) p p 2 and w p p (A) ≤ inf θ ∈ R Re ⁡ ((e i θ A) 2) p / 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 2 p / 2 for 2 ≤ p ≤ 4 , where ⋅ p is the Schatten p-norm. Applications of these inequalities to certain classes of operators are also given. [ABSTRACT FROM AUTHOR]

Let σ n (H) denote the class of operators A acting on a complex Hilbert space H such that W 0 (A) = c o (λ ∈ σ (A) : λ = A ) , where W 0 (A) and σ (A) are the maximal numericl range, the spectrum of A, respectively, and co stands for the convex hull. In this paper, we show that the class σ n (H) contains hyponormal operators. We also show that if two operators A ∈ σ n (H) and B ∈ σ n (H ′) have the same closure of the numerical range or have the same spectrum, then they also have the same maximal numerical range. An affirmative answer to the question 2.16 in Fialkow [Elementary operators and applications. Proceedings of the International Workshop; Bleuberen; 1991. p. 55–113] is also given. [ABSTRACT FROM AUTHOR]

A generalized Nash equilibrium problem corresponds to a noncooperative interaction between a finite set of players in which the cost function and the feasible set of each player depend on the decisions of the others. The classical existence result for generalized equilibria due to Arrow and Debreu requires continuity of the cost functions. In this work, we provide an existence of solutions transferring this hypothesis to a "continuity-like" condition over the sublevel sets of the aforementioned functions. Comparison with Reny's approach for discontinuous games is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we give an example of a densely defined non-closable paranormal operator. Then, we give another example of a densely defined closable paranormal operator whose closure fails to be paranormal. It is worth noticing that our first example says that the adjoint of a densely defined paranormal operator may have a trivial domain. In the end, we give an example of a densely defined closed one-to-one paranormal operator T whose adjoint is also injective and paranormal, but T is not normal.

In this paper, unitary symmetrizers are considered. It is well known that using Newton operator algorithm, similar to the usual Newton algorithm, for extracting the square root, one can prove that for every Hermitian operator T ≥ 0, there exists a unique Hermitian operator S ≥ 0 such that T = S². Moreover, S commutes with every bounded operator R with which commutes T. The operator S is called a square root of the operator T and is denoted by T1/2. The existence of the square root allows one to determine the absolute value |T| = (T*T)1/2 of the bounded operator T. For every bounded linear operator T : H → H there exists a unique partially isometric operator U : H → H such that T = U|T|, KerU = KerT. Such an equality is called a polar expansion of the operator T. The Schmidt operator is understood as the unitary multiplier of the polar expansion of a compact inverse operator, with the help of which E. Schmidt was the first to obtain the expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers. This paper shows that the unitary symmetrizer of an operator differs only in sign from the adjoint Schmidt operator. The main result of the paper: if A is an invertible and compact operator, and S is a unitary operator such that the operator SA is self-adjoint, then the operator AS is also self-adjoint and the formula S = ±U* holds, where U is the Schmidt operator. [ABSTRACT FROM AUTHOR]

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the x-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions. [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 introduce a new type of Drazin invertible operator on Hilbert spaces, which is called D-operator. Then, some properties of the class of Doperators are studied. We prove that the D-operator preserves the scalar product, the unitary equivalent property, the product and sum of two D-operators are not D-operator in general but the direct product and tenser product is also D-operator. [ABSTRACT FROM AUTHOR]

Some stronger and equivalent metrics are defined on $\mathcal{M}$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $\mathcal{M}$ are investigated.

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]

Let T be a bounded linear operator on a Hilbert space H. We say that T has the hyponormal property if there exists a function f, continuous on an appropriate set so that f(|T|) = f(|T*|). We investigate the properties of such operators considering certain classes of functions on which our definition is constructed. For such a function f we introduce the f-Aluthge transform, Tf. Given two continuous functions f and g with the property f(t)g(t) = t, we also introduce the (f, g)-Aluthge transform, T(f,g). The features of these transforms are discussed as well. [ABSTRACT FROM AUTHOR]

In this paper, we show that the numerical range of a bounded linear operator T on a complex Hilbert space is a line segment if and only if there are scalars λ and µ such that T∗ = λT + µI, and we determine the equation of the straight support of this numerical range in terms of λ and µ. An operator T is called sub-self-adjoint if their numerical range is a line segment. The class of sub-self-adjoint operators contains every self-adjoint operator and contained in the class of normal operators. We show that this class is uniformly closed, invariant under unitary equivalence and invariant under affine transformation. Some properties of the sub-self-adjoint operators and their numerical ranges are investigated. [ABSTRACT FROM AUTHOR]