We show that hyperrigidity for a C*-correspondence (A, X) is equivalent to non-degeneracy of the left action of the Katsura ideal \(\mathcal {J}_X\) on X. This extends the work of Kakariadis (Bull Lond Math Soc 45(6):1119–1130, 2013, Theorem 3.3) and Dor-On and Salomon (J Lond Math Soc 98(2):416–438, 2018, Theorem 3.5) who establish this equivalence for discrete graphs as well as the work of Katsoulis

We use some results from the theory of reproducing kernel Hilbert spaces to show that the reachable space of the heat equation for a finite rod with either one or two Dirichlet boundary controls is a RKHS of analytic functions on a square, and we compute its reproducing kernel as an infinite double series. We also show that the null reachable space of the heat equation for the half line with Dirichlet

By the linearization property of Lipschitz-free spaces, any Lipschitz map \(f : M \rightarrow N\) between two pointed metric spaces may be extended uniquely to a bounded linear operator \({\widehat{f}} : {\mathcal {F}}(M) \rightarrow {\mathcal {F}}(N)\) between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps \(f : M \rightarrow

We define the angle of a bounded linear operator A along a curve emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if \(\sigma (A)\) does not separate 0 from \(\infty \), then \(X=R(A)\oplus N(A)\) if and only if R(A) is closed and some angle of A is less than \(\pi \). We first apply this result to invertible operators that have a spectral

We establish the holomorphic dependence of the boundary integral operators (BIOs) comprising the Calderón projector for the Laplacian in two dimensions on the boundary shape. More precisely, we show that the Calderón projector, as an element of the Banach space of bounded linear operators satisfying suitable mapping properties, depends holomorphically on a set of boundaries given by a collection of

For oscillatory singular integrals with polynomial phases and Hölder class kernels, we establish their uniform boundedness on \(L^p\) spaces as well as a sharp logarithmic bound on the Hardy space \(H^1\). These results improve the ones in (Pan in Forum Math 31: 535–542, 2019) by removing the restriction that the phase polynomials be quadratic.

For an arbitrary operator T acting on a Hilbert space we consider a field of operators \(\left( \Delta _{z}(T)\right) \) called the Aluthge operator field associated with T. After giving preliminary results, we establish that two fields (left and right), canonically linked to the Altuthge field \(\left( \Delta _{z}(T)\right) \) and a support subspace, are constant on each horizontal segment where they

In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed

The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheless share many features with Schrödinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous

This paper is devoted to the multivariable \(H^\infty \) functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if \((A_1,\ldots , A_d)\) is such a family, if \(A_k\) is R-sectorial of R-type \(\omega _k\in (0,\pi )\), \(k=1,\ldots ,d\), and if \((A_1,\ldots , A_d)\) admits a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times

This work is a continuation of our previous paper (Zelenko in Appl Anal Optim 3(2):281–306, 2019), where for the Schrödinger operator \(H=-\Delta + V({{\mathbf {x}}})\cdot \), acting in the space \(L_2({{\mathbf {R}}}^d)\,(d\ge 3)\), some sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya–Shubin criterion and an optimization problem for a set function

The purpose of this paper is to revisit the proof of the Gearhart–Prüss–Huang–Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on \(\Vert S(t) \Vert \) in terms of bounds on the resolvent of the generator. A first version of this paper was presented by the two authors in ArXiv (2010) together with applications

Let \(\Omega \subset {\mathbb {C}}^n\) be a smooth bounded pseudoconvex domain and \(A^2 (\Omega )\) denote its Bergman space. Let \(P:L^2(\Omega )\longrightarrow A^2(\Omega )\) be the Bergman projection. For a measurable \(\varphi :\Omega \longrightarrow \Omega \), the projected composition operator is defined by \((K_\varphi f)(z) = P(f \circ \varphi )(z), z \in \Omega , f\in A^2 (\Omega ).\) In

An interesting result proved by Halmos in Hal (Michigan Mathematical Journal, 15, 215–223 (1968) is that the set of irreducible operators is dense in \({\mathcal {B}}({\mathcal {H}})\) in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra \({\mathcal {M}}\) with separable predual, an operator \(a\in {\mathcal {M}}\) is said to be irreducible in \({\mathcal {M}}\) if \(W^*(a)\) is

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell–Lauricella hypergeometric function of type

We prove the Cesàro boundedness of eigenfunctions of the Dirac operator on the half-line with a square-summable potential. The proof is based on the theory of Krein systems and, in particular, on the continuous version of a theorem by A. Máté, P. Nevai and V. Totik from 1991.

We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible

On the Euclidean space \({\mathbb {R}}^N\) equipped with a normalized root system R, a multiplicity function \(k\ge 0\), and the associated measure \(dw({\mathbf {x}})=\prod _{\alpha \in R} |\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}d{\mathbf {x}}\) we consider the differential-difference operator $$\begin{aligned} L=(-1)^{\ell +1} \sum _{j=1}^m T_{\zeta _j}^{2\ell }, \end{aligned}$$ where

Four new examples of explicitly diagonalizable Hankel matrices depending on a parameter \(k\in (0,1)\) are presented. The Hankel matrices are regarded as matrix operators on the Hilbert space \(\ell ^{2}(\mathbb {N}_{0})\) and the solution of the spectral problem is based on an application of the commutator method. Each of the Hankel matrices commutes with a Jacobi matrix which is related to a particular

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in Kennedy et al. (Calc Var 60:6, 2021). We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror

In this paper, we study the boundedness and the compactness of the little Hankel operators \(h_b\) with operator-valued symbols b between different weighted vector-valued Bergman spaces on the open unit ball \(\mathbb {B}_n\) in \(\mathbb {C}^n.\) More precisely, given two complex Banach spaces X, Y, and \(0 < p,q \le 1,\) we characterize those operator-valued symbols \(b: \mathbb {B}_{n}\rightarrow

For a quasinilpotent bounded linear operator T, we write \(k_x=\limsup \limits _{z\rightarrow 0}\frac{\log \Vert (z-T)^{-1}x\Vert }{\log \Vert (z-T)^{-1}\Vert }\) for each nonzero vector x. Set \(\Lambda (T)=\{k_x:x\ne 0\}\), and call it the power set \(\Lambda (T)\) of T. This notation was introduced by Douglas and Yang (see Hermitian geometry on resolvent set. arXiv:1608.05990, 2016; Oper Theory

We determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators on weighted Bergman spaces over bounded symmetric domains \(D=G/K,\) for the irreducible K-types indexed by all partitions of length \(r={\mathrm {rank}}(D)\).

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter \(\epsilon \) denotes the thickness of the domain and the roughness periodicity

Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular

In this article we solve four special cases of the truncated Hamburger moment problem (THMP) of degree 2k with one or two missing moments in the sequence. As corollaries we obtain via appropriate substitutions, the solutions to bivariate truncated moment problems of degree 2k for the curves \(y=x^3\) (first solved by Fialkow in Trans Am Math Soc 363:3133–3165, 2011), \(y^2=x^3\), \(y=x^4\) where a

The Hilbert matrix $$\begin{aligned} {\mathcal {H}}_\lambda =\left( \frac{1}{n+m+\lambda }\right) _{n,m=0}^{\infty }, \quad \lambda \ne 0,-1,-2, \ldots \, \end{aligned}$$ generates a bounded linear operator in the Hardy spaces \(H^p\) and in the \(l^p\)-spaces. The aim of this paper is to study the spectrum of this operator in the spaces mentioned. In a sense, the presented investigation continues

In this paper, we construct, for a certain class of semigroup dynamical systems, two operator algebras that are universal with respect to their corresponding covariance conditions: one being selfadjoint, and another being nonselfadjoint. We prove that the C*-envelope of the nonselfadjoint operator algebra is precisely the selfadjoint one. This result leads to a number of new examples of operator algebras

It is known that exactness for a discrete group is equivalent to C*-exactness, i.e., the exactness of its reduced C*-algebra. The problem of whether this equivalence holds for general locally compact groups has recently been reduced by Cave and Zacharias to the case of totally disconnected unimodular groups. We prove that the equivalence does hold for a class of groups that includes all locally compact

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 $$A_{\alpha }^p$$ A α p with $$p \in (0, \infty )$$

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

We identify the weighted $$L^p$$ 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 $g, Ag, A^2 g, \dots$ , and whether solutions

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 (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint

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 spaces are

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 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 $\Theta$ is a pure operator valued inner function

Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger 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 these

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 [17,19] where we studied the compactness of corresponding embeddings. The concept of nuclearity goes back to Grothendieck who defined it in [14]. Recently there is a refreshed interest to

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 Szego theorem for Bergman–Toeplitz matrices

In an earlier paper (A. N. Kochubei, {\it 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 under 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 of the

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