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Let E and F be two complex Hilbert spaces. Let S_{E} and S_{F} be the shift operators on vector-valued Hardy space H_{E}^{2} and H_{F}^{2}, respectively. We show that the cyclic multiplicity of S_{E}\oplus S_{F} ^{\ast } equals 1+\dim E. This result is classical when \dim E=\dim F=1 (see J. A. Deddens [ On A \oplus A^\ast, 1972]; Paul Richard Halmos [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos's book [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for S_{E}\oplus S_{F}^{\ast } (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of S_{E}\oplus S_{F}^{\ast } when \dim E<\infty and \dim F=1. [ABSTRACT FROM AUTHOR]

A criterion for a contraction T on a Hilbert space to be complex symmetric is given in terms of the operator-valued characteristic function ΘT of T in 2007 (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]). To further classify unitary equivalent complex symmetric contractions, we notice a simple condition of when ΘT1 and ΘT2 coincide for two complex symmetric contractions T1 and T2. As an application, surprisingly we solve the problem for any defect index n, when the defect indexes of contractions are 2, this problem was left open by Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]. Furthermore, a construction of 3 × 3 symmetric inner matrices is proposed, which extends some results on 2 × 2 inner matrices (see Stephan Ramon Garcia [J. Operator Theory 54 (2005), pp. 239–250]) and 2 × 2 symmetric inner matrices (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]). [ABSTRACT FROM AUTHOR]

We describe all the invariant subspaces of a completely nonunitary contraction T_c with nonzero scalar constant characteristic function c, where |c|<1. Moreover, we show that T_c and restriction to its invariant subspaces are essentially normal. Using this description of T_c-invariant subspaces, we then identify the hyperinvariant subspaces. [ABSTRACT FROM AUTHOR]

The spectral mapping theorem, in conjunction with the concept of a spectral set, is utilized to construct reducing pairs of subspaces for a given operator. By leveraging the theory of functional calculus, we revisit the well-established spectral mapping theorem for various spectra, with a particular focus on the point spectrum. Concluding this discussion, we present an application wherein we collect and establish properties of the point spectrum of a compact exponentially m-isometry. Also, we examine the behavior of solutions of a system of differential equations associated with exponentially m-isometric matrices. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to introduce the twisted noncommutative Hardy algebra F^\infty (\mathcal {U}, \mathbf {S}) associated with the standard \mathcal {U}-twisted multi-shift \mathbf {S}, describe some of its basic properties, and develop a w^*-continuous functional calculus for the pure (resp. completely non-coisometric) elements in the regular \mathcal {U}-twisted polyball \mathbf {B}_\mathcal {U}(\mathcal {H}). In addition, we introduce the twisted noncommutative polyball algebra \mathcal {A}(\mathcal {U}, \mathbf {S}) and present the corresponding functional calculus for the elements in \mathbf {B}_\mathcal {U}(\mathcal {H}). [ABSTRACT FROM AUTHOR]

We present a quick introduction to Frobenius complexity and gradings with rational twist for commutative rings of prime characteristic, providing a place to start for those interested in exploring these topics. [ABSTRACT FROM AUTHOR]

In this article, we survey recent results concerning non-stability of discrete groups with respect to the operator norm. We focus on topological obstructions to perturbing almost representations of a discrete group r into unitary groups U(n) to true representations. Several natural notions of stability are discussed: local-to-local stability, uniform-to-uniform stability, uniform-to-local stability, and C * -stability. [ABSTRACT FROM AUTHOR]

We continue the study of the perturbation problem discussed in H. D. Cornean and R. Purice (2023) and get rid of the "slow variation" assumption by considering symbols of the form a(x+δ F(x), ξ) with α a real Hörmander symbol of class S 0,00(ℝd x ℝd) and F a smooth function with all its derivatives globally bounded, with |δ| ≤ 1. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of δ = 0 is still of the order √|δ|, the distance between their spectral edges behaves like |δ|ν with ν ϵ [1/2,1) depending on the rate of decay of the second derivatives of F at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the strongly convergent Krasnoselskii-Mann-type iteration for families of nonexpansive mappings defined recently by Bot and Meier in Hilbert spaces to the abstract setting of W-hyperbolic spaces and we compute effective rates of asymptotic regularity for our generalization. This also extends recent results by Leustean and the author on the Tikhonov-Mann iteration from single mappings to families of mappings. [ABSTRACT FROM AUTHOR]

We present a technique that can be used in the study of a conjecture by Danilov and Koshevoy, concerning triples (A, B,C) of n x n complex selfadjoint matrices such that C = A + B. The conjecture proposes an explicit formula, in terms of traces of compressions of A and B, for one associated hive. We also use this technique to show why an earlier attempt to prove the conjecture fails for n = 4. AMS 2020 Subject Classification: 15A18. [ABSTRACT FROM AUTHOR]

For a Banach algebra A, we say that an element M in A ⊗γ A is a hypercommutator if (a ⊗ 1)M = M(1 ⊗ a) for every a ∈ A. A diagonal for a Banach algebra is a hyper-commutator whose image under diagonal mapping is 1. It is well known that a Banach algebra is contractible iff it has a diagonal. The main aim of this note is to show that for any Banach subalgebra A ⊆ L(X) of bounded linear operators on infinite-dimensional Banach space X, which contains the ideal of finite-rank operators, the image of any hyper-commutator of A under the canonical algebra-morphism L(X) ⊗γ L(X) → L(X ⊗γ X), vanishes. [ABSTRACT FROM AUTHOR]

We present an alternative construction – using forcing, instead of diamond principle – of a Hausdorff compactum K such that C(L) has few operators, for every closed L ⊂ K. Moreover, we prove a new result about the density of C(K) with few operators and present some open problems. [ABSTRACT FROM AUTHOR]

Algebra environments capture properties of non-commutative conditional expectations in a general algebraic setting. Their study relies on algebraic geometry, topology, and differential geometry techniques. The structure algebraic and Banach manifolds of algebra environments and their Zariski and smooth tangent vector bundles are particular objects of interest. A description of derivations on algebra environments compatible with geometric structures is an additional issue. Grassmann and flag manifolds of unital involutive algebras and spaces of projective compact group representations in C* -algebras are analyzed as structure manifolds of associated algebra environments. [ABSTRACT FROM AUTHOR]

We discuss the concept of decomposability for operators from its inception to present applications. [ABSTRACT FROM AUTHOR]

We show that truncated Toeplitz operators are characterized by a collection of complex symmetries. This was conjectured by Kliś-Garlicka, Lanucha, and Ptak, and proved by them in some special cases. [ABSTRACT FROM AUTHOR]

We present a class of m-isometries on a Hilbert space which admit Wold-type decompositions in Shimorin’s sense. Among these operators, we recover some sub-Brownian m-isometries and theirs m-Brownian unitary extensions. Our context refers to an integer m ≥ 3, the cases m = 1 and m = 2 being well-known and studied. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey Triebel-Lizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider little Hankel operators Γφ defined on the Bergman space L²a(D) with symbol φ ∈ H∞(D) that are contractions. Necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part of these little Hankel operators. We also present an explicit description of this unitary part. This extends the results of Butz for Hankel operators defined on the Hardy space. [ABSTRACT FROM AUTHOR]

We consider the orthogonality preserving property to study sesquilinear forms. We state some characterizations for Hermitian sesquilinear forms through this kind of approach. We, then, state a Wigner type equation and present some results in this regard. We investigate the superstability of the perturbation of such equation. Finally, we state some results in inner product C*-modules concerning the issue. [ABSTRACT FROM AUTHOR]

The Tenth Congress of Romanian Mathematicians was held in Pites, Romania, from June 30th to July 5th, 2023, with over 380 participants from 41 countries in attendance. The congress, organized by various institutions, featured eight parallel sections covering a wide range of mathematical topics, including algebra, geometry, analysis, differential equations, and theoretical computer science. Additionally, the congress hosted the first Balkan Mathematics Conference, and this volume contains 16 articles by speakers at the event. [Extracted from the article]

Due to their potential in pure and applied mathematics, the notions of basis and frame have various generalizations. Recently, g-frame and g-basis were introduced and studied by some mathematicians. It is well known that a frame-based series expansion of a vector is unconditionally convergent, while a basis-based one need not be. In applications unconditionality is more favourable than conditionality. In this paper, we introduce the notion of g-unconditional basis which leads to unconditional convergence, and establish a characterization of g-unconditional bases. [ABSTRACT FROM AUTHOR]

Let (A0, A1) be an interpolation couple such that A0 is a subspace of A1 and the identity i : A0 → A1 is continuous. In the second part of this work, we show that if A1 is weakly sequentially complete, then Aϑ,p has the same property for ϑ ∈ ]0,1[ and p ∈ ]1, + ∞[. We give an example for which the previous conclusion is false if the assumption on A1 is replaced by the same one on A0. In the third part, we show that there is an interpolation couple (A0, A1) such that A0 has the Radon-Nikodym property, A0 is a subspace of A1, and (L(L¹, A0), L(L¹, A1))ϑ is a strict isometric subspace of C(L¹,Aϑ), ϑ ∈ ]0,1[. Finally, we show in this part, that if i : A0 → A1 is weakly compact, then Aϑ+ is a dual space. [ABSTRACT FROM AUTHOR]

Motivated by recent progress in operator representation of frames, we investigate the frames of the form {Tnφ}n∈I for I = N, Z, and answer questions about representations, perturbations and frames induced by the action of powers of bounded linear operators. As a particular case, we discuss problems concerning representation of frames in terms of iterations of the mixed frame operators. As our another contribution, we consider frames of the form {anTnφ}∞n=0 for some non-zero scalars {an}∞n=0, and we obtain some new results in dynamical sampling. Finally, we will present some auxiliary results related to the perturbation of sequences of the form {Tnφ}∞n=0. [ABSTRACT FROM AUTHOR]

In this paper, we present some upper and lower bounds for operator space numerical radius of k×k block matrices, with entries inMn(X), when X is a numerical radius operator space. We also derive formulas for operator space numerical radius of some special bidiagonal block matrices. One of our main results states that if (X, (Wn)) is a numerical radius operator space and x1, . . ., xk ?Mn(X), then*** where ω = e 2πi/k is the k-th root of unity. [ABSTRACT FROM AUTHOR]

Using an explicit construction, we show that the de Branges–Rovnyak spaces H (b) , corresponding to rational b, are contained in their associated Smirnov classes. [ABSTRACT FROM AUTHOR]

Given a strictly increasing sequence ʌ=(λn) of nonnegative real numbers, with ..., theMüntz spaces Mʌp. are defined as the closure in Lp([0, 1]) of the monomials xλn. We discuss properties of the embedding Mʌp⊂Lp(μ), where μ is a finite positive Borel measure on the interval [0, 1]. Most of the results are obtained for the Hilbertian case p = 2, in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten-von Neumann ideals. [ABSTRACT FROM AUTHOR]

Consider the following problem: given complex numbers a1, …, an, find an L∞ function f of minimum norm whose Fourier coefficients ck(f) are equal to ak for k between 0 and n. We show the uniqueness of this function, and we estimate its norm. The operator-valued case is also discussed. 2000 Mathematics Subject Classification 30E05, 47A20, 47A56, 47A57. [ABSTRACT FROM PUBLISHER]

Given a contraction A on a Hilbert space H , an operator T on H is said to be A-invariant if ⟨ T x , x ⟩ = ⟨ T A x , A x ⟩ for every x ∈ H such that ‖ A x ‖ = ‖ x ‖ . In the special case in which both defect indices of A are equal to 1, we show that every A-invariant operator is the compression to H of an unbounded linear transformation that commutes with the minimal unitary dilation of A. This result was proved by Sarason under the additional hypothesis that A is of class C 00 , leading to an intrinsic characterization of the truncated Toeplitz operators. We also adapt to our more general context other results about truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra A . After obtaining some general results, we classify all the maximal algebras for certain particular cases of A . [ABSTRACT FROM AUTHOR]

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincaré operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey. [ABSTRACT FROM AUTHOR]