Hilbert Space Operators Research Articles

Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type.

Norm-Attainable Operators in Hilbert Spaces: Properties and Applications

This research paper explores various properties and implications of norm-attainable operators on Hilbert spaces. We establish lemmas, propositions, and theorems that shed light on the characteristics of these operators and their relationship with the geometry and structure of the underlying Hilbert space. These results have applications in functional analysis, linear algebra, and operator theory.

A note on non-supercyclic vectors of Hilbert space operators

In this note it is shown that there is a bounded linear operator T on the Hardy Hilbert space H2 and a vector f in H2 such that the closure of the set {αTnf:α∈ℂ,n≥0} is not H2, but for every subsequence (nk)k=1∞ the closed linear span of {Tnkf:k≥1} is the whole space H2. Furthermore, the closure of {Tng:n≥0} is H2 for some g∈H2.

Upper bounds for the Euclidean spectral radius of operators via joint norms

Let r e ( S ) be the Euclidean spectral radius associated with a q-tuple S = ( S 1 , … , S q ) of bounded linear operators on a complex Hilbert space. The principal objective of our study is to establish various compelling upper bounds involving r e ( ⋅ ) . In particular, our findings demonstrate that, for all t ∈ [ 0 , 1 ] , we have r e ( S ) ≤ 1 2 max { ω ( S ) , ω ( Δ t ( S ) ) } + 1 2 ‖ S ‖ 1 2 ‖ S ‖ e 1 2 . Here, Δ t ( S ) represents the generalized spherical Aluthge transform of S , while the notations ω ( ⋅ ) , ‖ ⋅ ‖ , and ‖ ⋅ ‖ e pertain to the joint numerical radius, joint operator norm, and Euclidean operator norm, respectively, of operators in Hilbert spaces. Furthermore, we extend the notions of spherical and Duggal transforms and derive multiple upper bounds for r e ( S ) in relation to these transforms. Additionally, there are some applications that are derived as well.

Numerical radii of operator matrices in terms of certain complex combinations of operators

Abstract Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.

Core inverse of operators in Hilbert spaces

Core inverse of operators in Hilbert spaces

Inequalities for power series of product of operators in Hilbert spaces with applications to numerical radius

Inequalities for power series of product of operators in Hilbert spaces with applications to numerical radius

Alternated inertial forward-backward-forward splitting algorithm

In this paper, we propose a forward-backward-forward splitting algorithm with alternated inertial step for finding a zero of the sum of two monotone operators in a real Hilbert space. Then, by making use of primal-dual techniques we derive the alternated inertial primal-dual splitting algorithm of forward-backward-forward type for solving structured monotone inclusion problems involving parallel sums and compositions of maximally monotone operators with linear continuous ones and convex minimization problems. Finally, we demonstrate numerically that our proposed algorithm performs better than the existing algorithms.

Novel bounds for the Euclidean operator radius of Hilbert space operator pairs

Novel bounds for the Euclidean operator radius of Hilbert space operator pairs

Some improvements about numerical radius inequalities for Hilbert space operators

Some improvements about numerical radius inequalities for Hilbert space operators

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.

Symmetric and antisymmetric tensor products for the function-theoretic operator theorist

Abstract We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

On tensor products with C-normal operators

A bounded linear operator A on H is said to be conjugate normal if CA⁎AC=AA⁎ for some conjugation C on H (in this case, A is said to be C-normal). In this paper, we investigate when the conjugate normality of an operator can be preserved under the operation of tensor product. Given an operator A, we show that the tensor product A⊗B is conjugate normal for any Hilbert space operator B if and only if A2=0. Also we obtain similar results concerning tensor products with skew symmetric operators, g-normal operators and Z-normal operators.

A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Raznostnye skhemy dekompozitsii na osnove rasshchepleniya resheniya i operatora zadachi

Domain decomposition methods are used for the approximate solution of boundary value problems for partial differential equations on parallel computing systems. The specifics of nonstationary problems is most completely taken into account when using noniterative domain decomposition schemes. Regionally additive schemes are constructed on the basis of various classes of splitting schemes. A new class of domain decomposition schemes with an additive representation of the solution on a new time level is distinguished that is based on splitting the domain into subdomains based on a partition of unity. An example of the Cauchy problem for first-order evolution equations with a positive self-adjoint operator in a finite-dimensional Hilbert space is considered. Unconditionally stable two- and three-level splitting schemes are constructed for the corresponding system of equations.

Two regularized inertial Tseng methods for solving inclusion problems with applications to convex bilevel programming

Combining the inertial technique, regularization, and the Tseng method for approximating a zero of the sum of monotone operators in a real Hilbert space, we propose two regularized inertial Tseng methods for approximating a zero of the sum of two maximal monotone operators in the case where one of them is monotone and Lipschitz continuous. We establish strong convergence of the sequences generated by our algorithms and derive several interesting consequences of our results. We then apply our results to solving convex bilevel programming problems and illustrate the cost efficiency of our methods by numerically comparing them with other methods in the literature.

Tensorial and Hadamard product inequalities for functions of selfadjoint operators in Hilbert spaces in terms of Kantorovich ratio

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with 0 <γ ≤ f(t)/g(t)≤ Γ for t ∈ I and if A and B are selfadjoint operators with Sp (A), Sp (B) ⊂ I, then [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)] ≤ (1 − ν) f (A) ⊗ g (B) + ν g (A) ⊗ f (B) ≤ [(γ + Γ) 2/4γΓ]R [f1−ν(A) gν (A)] ⊗ [fν(B) g1−ν (B)]. The above inequalities also hold for the Hadamard product “ ◦ ” instead of tensorial product “ ⊗ ”.

The Convergence of Some Spectral Characteristics on the Convergent Series

In this study, convergence properties of spectral, numerical and Crawford gap functions via convergences of Hilbert space operator series in difference and ratio cases are investigated. Obtained results have been applied to some classes continuous functions of the operators.

Generalized Euclidean operator radius

Abstract In this paper, we introduce the f-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces

In this paper several tensorial norm inequalities of Ostrowski type for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f.