For a given self-adjoint operator A with discrete spectrum, we completely characterise possible eigenvalues of its rank-one perturbations B and discuss the inverse problem of reconstructing B from its spectrum.

In the setting of the unit disk we have recently obtained characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators acting from a standard weighted Bergman space into the corresponding weighted Lebesgue space. In this paper we extend those results to the case when the exponents of the domain space and the target space are different.

Let \(X(\mathbb {R})\) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). The algebra \(C_X(\mathbf{\dot{\mathbb {R}}})\) of continuous Fourier multipliers on \(X(\mathbb {R})\) is defined as the closure of the set of continuous functions of bounded variation on \(\mathbf{\dot{\mathbb

This paper aims to study when a Toeplitz operator \(T_\varphi \) on the Hardy space \(H^2\) of the unit disk is complex symmetric, that is, \(T_\varphi \) admits a symmetric matrix representation relative to some orthonormal basis of \(H^2\). For certain trigonometric symbols \(\varphi \), we give necessary and sufficient conditions for \(T_\varphi \) to be complex symmetric. In particular, we show

For spaces of analytic functions defined on an open set in \({\mathbb {C}}^n\) that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space \(H^p({\mathbb {D}}^n) \, (0< p < \infty )\), the Drury–Arveson space \({\mathcal {H}}^2_n\), and

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain \(\Omega \subseteq \mathbb {R}^d\) with Lipschitz boundary \(\Gamma \). More precisely, using form methods, we show that the associated operator on the ground space \(L^2(\Omega )\times L^2(\Gamma )\) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators

Let \(B({{\mathcal {H}}})\) be the algebra of bounded linear operators on a Hilbert space \({{\mathcal {H}}}\) and consider \(\mathbf{D}_{\mathbf{f}}^{\mathbf{m}}({{\mathcal {H}}})\subset B({{\mathcal {H}}})^{n_1}\times \cdots \times B({{\mathcal {H}}})^{n_k}\) to be a regular polydomain generated by a k-tuple \(\mathbf{f}\,{{:}=}\,(f_1,\ldots , f_k)\) of positive regular free holomorphic functions

We consider dual Toeplitz operators acting on the orthogonal complements of the Fock spaces over the higher dimensional complex space. We then consider operators which are finite sums of products of two dual Toeplitz operators and study the problems of when such an operator is compact and zero respectively.

We prove a basic inequality involving anticommutators in noncommutative \(L_p\)-spaces. We use it to complete our study of the noncommutative Mazur maps from \(L_p\) to \(L_q\) showing that they are Lipschitz on balls when \(0

By means of a weaker functional model, we prove the existence of non-trivial closed hyperinvariant subspaces for linear bounded operators generalizing, in particular, a classical theorem of Atzmon and revealing the spectral nature of the hyperinvariant subspaces involved. As an application, we show non-trivial spectral subspaces for Bishop operators on \(L^p[0,1)\), \(1\le p<\infty \), as long as they

The paper deals with bounded linear operators T on a complex Hilbert space \({\mathcal {H}}\) which have 2-isometric liftings S on a Hilbert space \({\mathcal {K}}\) containing \({\mathcal {H}}\) as a closed subspace. We investigate three types of such liftings, and for each type we describe in detail the corresponding operators T by the \(2\times 2\) upper triangular block matrices. The entries of

Toeplitz matrices are typically non-Hermitian and hence they evade the well-elaborated machinery one can employ in the Hermitian case. In a pioneering paper of 1960, Palle Schmidt and Frank Spitzer showed that the eigenvalues of large banded Toeplitz matrices cluster along a certain limiting set which is the union of finitely many closed analytic arcs. Finding this limiting set nevertheless remains

A new method to enclose the pseudospectrum via the numerical range of the inverse of a matrix or linear operator is presented. The method is applied to finite-dimensional discretizations of an operator on an infinite-dimensional Hilbert space, and convergence results for different approximation schemes are obtained, including finite element methods. We show that the pseudospectrum of the full operator

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