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.

Let \({\mathcal {M}}\) be an arbitrary collection of linear-fractional non-automorphism self-maps of the unit disk \(\mathbb {D}\). We consider the unital \(\hbox {C}^*\)-algebras \(C^*(T_z, \{C_{\varphi } : \varphi \in {\mathcal {M}}\})\) and \(C^*(\{C_{\varphi } : \varphi \in {\mathcal {M}}\}, {\mathcal {K}})\) generated by the composition operators induced by the maps in \({\mathcal {M}}\) and either