The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces \(A_{\alpha }^p\) with \(p \in (0, \infty )\) and \(

In this note we investigate the essential normality of the Beurling-type quotient module \({\mathcal {D}}:=H^2(\Omega )\ominus \eta H^2(\Omega )\) with an inner function \(\eta \) inside \(A(\Omega )\) over an irreducible tube-type domain \(\Omega \). For the Lie ball (of rank 2), we characterize the essential normality of the corresponding quotient Hardy module and determine its essential spectrum

We identify the weighted \(L^p\)-norms of shift operators in the context of nonatomic probability spaces equipped with tree-like structures.

In this paper we consider some faithful representations of positive Hilbert space operators on structures of nonnegative real functions defined on the unit sphere of the Hilbert space in question. Those representations turn order relations between positive operators to order relations between real functions. Two of them turn the usual Löwner order between operators to the pointwise order between functions

The abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem \(Af=g\) where A is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of \(g,Ag,A^2g,\dots \), and whether solutions of this

In the original version of this article, the $ symbols were misinterpreted as part of the article text and placed in the article as it is.

We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections.

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar–Gallardo–Partington. Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting

We introduce the notion of \(\Delta \) and \(\sigma \,\Delta -\) pairs for operator algebras and characterise \(\Delta -\) pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of \(\Delta \)-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that \(\sigma \Delta \)-Morita equivalent operator

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations,

We extend Carleson’s interpolation theorem to sequences of matrices, by giving necessary and sufficient separation conditions for a sequence of matrices to be interpolating.

Conjugations commuting with \(\mathbf {M}_z\) and intertwining \(\mathbf {M}_z\) and \(\mathbf {M}_{{\bar{z}}}\) in \(L^2(\mathcal {H})\), where \(\mathcal {H}\) is a separable Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space \(H^2(\mathcal {H})\) or a model space \(K_\Theta =H^2(\mathcal {H})\ominus \Theta H^2(\mathcal {H})\), where \(\Theta

Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schrödinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional damped wave operator is similar to the undamped one provided that the \(L^1\) norm of the (possibly complex-valued) damping is less than 2. It follows that

We study nuclear embeddings for weighted spaces of Besov and Triebel–Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results concerning the compactness of corresponding embeddings. The concept of nuclearity was introduced by A. Grothendieck in 1955. Recently there is a refreshed interest to study such questions

In this paper, we study the asymptotic behavior of the determinants of Bergman–Toeplitz matrices with symbols in \(H^{\infty }({\mathbb {D}})+C(\overline{{\mathbb {D}}})\). We establish a criterion of the asymptotic invertibility and an asymptotic inversion formula for Bergman–Toeplitz operators. These results are applied to obtain two versions of the first Szegö theorem for Bergman–Toeplitz matrices

In an earlier paper (A. N. Kochubei, Pacif. J. Math. 269 (2014), 355–369), the author considered a restriction of Vladimirov’s fractional differentiation operator \(D^\alpha \), \(\alpha >0\), to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse \(I^\alpha \) that the appropriate change of variables reduces equations with \(D^\alpha \) (for radial

This paper concerns the final value problem for the heat equation subjected to the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work

Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which

Generalized multipliers for a left-invertible operator T, whose formal Laurent series \(U_x(z)=\sum _{n=1}^\infty (P_ET^{n}x)\frac{1}{z^n}+\sum _{n=0}^\infty (P_E{T^{\prime *n}}x)z^n\), \(x\in \mathcal {H}\) actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators

It was recently shown in Ter Horst et al. (Bull Lond Math Soc 51:1005–1014, 2019) that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular

This paper presents the first part of a study of functional models of selfadjoint and nonselfadjoint extensions \(\widetilde{A}\) of symmetric and nonsymmetric operators A in a Hilbert space \(\mathfrak {H}\). The extensions will be considered in the framework of linear relations (which may also be interpreted as the graphs of multivalued operators) that are required to have a nonempty set of regular

The techniques developed by Popescu, Muhly-Solel and Good for the study of algebras generated by weighted shifts are applied to generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning dilations and invariant subspaces for commuting tuples of operators. In that paper the authors prove Beurling-Lax-Halmos type results for commuting tuples \(T=(T_1,\ldots ,T_d)\) of operators

A characterisation is given of bounded embeddings from weighted \(L^2\) spaces on bounded intervals into \(L^2\) spaces on the half-plane, induced by isomorphisms given by the Laplace transform onto weighted Hardy and Bergman spaces (Zen spaces). As an application necessary and sufficient conditions are given for the boundedness of truncated Hankel and Toeplitz integral operators, including the weighted

Let A, T, and B be bounded linear operators on a Banach space. This paper is concerned mainly with finding some necessary and sufficient conditions for convergence in operator norm of the sequences \(\left\{ A^{n}TB^{n}\right\} \) and \(\left\{ \frac{1}{n}\sum _{i=0}^{n-1}A^{i}TB^{i} \right\} \). These results are applied to the Toeplitz, composition, and model operators. Some related problems are

It is well known the kernel of a Toeplitz operator is nearly invariant under the backward shift \(S^*\). This paper shows that kernels of finite rank perturbations of Toeplitz operators are nearly \(S^*\)-invariant with finite defect. This enables us to apply a recent theorem by Chalendar–Gallardo–Partington to represent the kernel in terms of backward shift-invariant subspaces, which we identify in

We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and Hörmander algebras of entire functions.

In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore

The paper establishes arithmetic–geometric mean and related submajorisation and norm inequalities for \(\tau \)-measurable operators affiliated with a semi-finite von Neumann algebra acting in a separable Hilbert space as an application of the theory of double operator integration.

We show that convolution operators on certain spaces of entire functions of a given type and order on Banach spaces are strongly mixing with respect to an invariant Borel probability measure with full support (a stronger property than frequent hypercyclicity). Based on results of S. Muro, D. Pinasco and M. Savransky we also show the existence of frequently hypercyclic entire functions of exponential

In this paper, we consider the spectral problem for a Stieltjes string with both ends fixed and with one-dimensional damping at an interior point. We solve an inverse problem of recovering parameters of the Stieltjes string using partial information on the string and partial information on the spectrum. First we prove uniqueness and give an algorithm of recovering the unknown parameters of the string

In the present note a spectral theorem for a finite tuple of pairwise commuting, self-adjoint and definitizable bounded linear operators \(A_1,\ldots ,A_n\) on a Krein space is derived by developing a functional calculus \(\phi \mapsto \phi (A_1,\ldots ,A_n)\) which is the proper analogue of \(\phi \mapsto \int \phi \, dE\) in the Hilbert space situation with the common spectral measure E for a finite

The paper establishes arithmetic-geometric mean and related submajorisation and norm inequalities in the general setting of \(\tau \)-measurable operators affiliated with a semi-finite von Neumann algebra.

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in \({\mathbb {C}}^n\). The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman

We establish Ambrosetti–Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic

The classical result by Brown and Halmos (J Reine Angew Math 213:8–102, 1964) implies that there is no nontrivial commutative \(C^*\)-algebra generated by Toeplitz operators acting on the Hardy space \(H^2(S^1)\), while there are only two commutative Banach algebras. One of them is generated by Toeplitz operators with analytic symbols, and the other one is generated by Toeplitz operators with conjugate

We consider an inverse problem for Schrödinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be described by the roots of a polynomial. We verify that one of the roots is equal to the mean value of the potential and apply it to prove an Ambarzumian type result

Sturm–Liouville problems on simple connected equilateral graphs of \(\le 5\) vertices and trees of \(\le 8\) vertices are considered with Kirchhoff’s and continuity conditions at the interior vertices and Neumann conditions at the pendant vertices and the same potential on the edges. It is proved that if the spectrum of such problem is unperturbed (such as in case of zero potential) then this spectrum

In this paper, we characterize bounded Toeplitz product \(T_fT_{g}\) and Hankel product \(H_f^{*}H_g\) on Fock space \(F_{\alpha }^2\) for two polynomials f and g in z and \({\overline{z}}\). As a consequence, we obtain when Toeplitz operator \(T_f\) or Hankel operator \(H_f\) with the polynomial symbol f in z and \({\overline{z}}\) is bounded.

This paper provides decay bounds for Green matrices and generalized eigenvectors of block Jacobi operators when the real part of the spectral parameter lies in a bounded gap of the operator’s essential spectrum. The case of the spectral parameter being an eigenvalue is also considered. It is also shown that if the matrix entries commute, then the estimates can be refined. Finally, various examples

We introduce the space of analytic functions on the open unit disk whose derivative has the p-th integrable bounded characteristic on the unit circle. We will give the characterization of a linear isometry of this space. We also characterize the surjective, not necessarily linear, multiplicative isometry of this space.

We generalize results of Davie and Raeburn describing homotopy types of the group of invertible elements and of the set of idempotents of the projective tensor product of complex unital Banach algebras. We illustrate our results by specific examples.

Let \({\mathbb M}\) be the complexification of the quaternionic algebra \({\mathbb H}\). For each function \(F:U\mapsto {\mathbb M}\), where \(U\subset {\mathbb C}\), we define a transformation \(F_{\mathbb H}:U_{\mathbb H}\mapsto {\mathbb M}\), where \(U_{\mathbb H}\subset {\mathbb H}\) is associated to U, via an elementary functional calculus, using the spectra of quaternions, and characterize those

We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for Rayleigh-Wood

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \({\mathsf {H}}\) on an unbounded, radially symmetric (generalized) parabolic layer \({\mathcal {P}}\subset {\mathbb {R}}^3\). It was known before that \({\mathsf {H}}\) has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics

We show that in a scale of weighted Dirichlet spaces \(D_{\alpha }\), including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in \(D_{\alpha }\) such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance,

For a countable discrete group \(\Gamma \) and a minimal \(\Gamma \)-space X, we study the notion of \((\Gamma , X)\)-boundary, which is a natural generalization of the notion of topological \(\Gamma \)-boundary in the sense of Furstenberg. We give characterizations of the \((\Gamma , X)\)-boundary in terms of essential or proximal extensions. The characterization is used to answer a problem of Hadwin

We determine the norm of a weighted composition operator \(W_{\psi ,\varphi }\), acting on the Hardy space \(H^{2}\) or one of the weighted Bergman spaces \(A_{\alpha }^{2}\), in the case where the composition symbol \(\varphi \) is an automorphism of the unit disk. Furthermore, we characterize all such operators that have maximal norm relative to an upper bound stated in terms of \(\Vert \psi \Vert

Analytic Morrey spaces belong to the class of function spaces which, like BMOA, are defined in terms of the degree of oscillation on the boundary of functions analytic in the unit disc. We consider semigroups of composition operators on these spaces and focus on the question of strong continuity. It is shown that these semigroups behave like on BMOA.

We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide uniform asymptotics of their generalised eigenvectors. We illustrate our results by considering complex perturbations of real Jacobi matrices belonging to several classes:

The issue discussed in this paper is: how small can a sum of idempotents be? Here smallness is understood in terms of nilpotency or quasinilpotency. Thus the question is: given idempotents \(p_1,\ldots ,p_n\) in a complex algebra or Banach algebra, is it possible that their sum \(p_1+\cdots +p_n\) is quasinilpotent or (even) nilpotent (of a certain order)? The motivation for considering this problem

We construct inverses of the variational electric field boundary integral operator up to compact perturbations on orientable topologically simple screens. We describe them as solution operators of variational problems set in low-regularity standard trace spaces. On flat disks these variational problems do not involve the inversion of any non-local operators. This result lays the foundation for operator

The subnormal completion problem on a directed tree is to determine, given a collection of weights on a subtree, whether the weights may be completed to the weights of a subnormal weighted shift on the directed tree. We study this problem on a directed tree with a single branching point, \(\eta \) branches and the trunk of length 1 and its subtree which is the “truncation” of the full tree to vertices

We present direct and self-contained proofs of the existence of singular traces on sufficiently large classes of operator ideals that live on the separable infinite-dimensional real Hilbert space. Our only tool is Banach’s version of the extension theorem of linear forms. Thanks to the use of dyadic representations of operators, all proofs have become straightforward.

We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its

In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences \(q_n\) and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index

We look for generalizations of the Brown–Douglas–Fillmore essential codimension result, leading to interesting local uniqueness theorems in KK theory. We also study the structure of Paschke dual algebras.

We initiate the study of the forward and backward shifts on the Lipschitz space of an undirected tree, \(\mathcal {L}\), and on the little Lipschitz space of an undirected tree, \(\mathcal {L}_0\). We determine that the forward shift is bounded both on \(\mathcal {L}\) and on \(\mathcal {L}_0\) and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift

Motivated by results in uniform algebras, a distance localization formula in C*-algebras is established in the framework of the Allan–Douglas localization principle, and is used to derive a locality result for products of Hankel operators as compact perturbations of Hankel operators. Using localization and certain QC functions, it is proved that the essential spectrum of the commutator of a Toeplitz

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the \(N\times N\) truncated Hilbert matrix for large values of N. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g.

We study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schrödinger