In this paper, we introduce and investigate the notion of the Ces & aacute;ro operator CT on a rooted directed tree T. When T is the rooted tree with no branching vertex, then CT is unitarily equivalent to the classical Ces & aacute;ro operator C0 on the sequence space & ell;2(N). We prove that for every narrow rooted directed tree T, CT is bounded, with norm bounded above by twice the square root of the width of T. When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming T is leafless and narrow, we show that CT is subnormal if and only if T is isomorphic to the rooted directed tree without any branching vertex. In particular, this demonstrates that the verbatim analogue of Kriete-Trutt theorem fails in the context of Ces & aacute;ro operators on rooted directed trees. Nonetheless, under the same hypotheses, CT is always a compact perturbation of a subnormal operator.

We discuss the boundedness, Schatten-class properties and scattering theory of Helson matrices. We also discuss a class of Helson matrices induced by positive and signed measures. All the results of this paper are illustrated with several examples not considered earlier.

Let S lambda be a bounded left-invertible weighted shift on a rootless directed tree T = (V, E). We address the question of when S lambda has Wold-type decomposition. We relate this problem to the convergence of the series X infinity n=1 X u is an element of Gv,n\Gv,n-1 C lambda(n) (u)2 , v is an element of V, lambda(n)(v) involving the moments lambda(n) of S lambda & lowast;, where Gv,n = Chi < n >(par < n >(v)). Our main result characterizes all bounded left-invertible weighted shifts S lambda on T which have Wold-type decomposition.

The Szegö-Dirichlet kernel of the right half-plane ℍ_1/2 is given by ϰ(s, u) = ζ (s+u), s, u ∈ℍ_1/2, where ζ denotes the Riemann zeta function. We show that none of the positive integer powers of ϰ has the 2-point scalar Pick property. Nevertheless, a network realization formula for the right half-plane ℍ_0 is obtained.

Let σ : ℂ^d →ℂ^d be an affine-linear involution such that J_σ = -1 and let U, V be two domains in ℂ^d. Let ϕ : U → V be a σ -invariant 2-proper map such that J_ϕ is affine-linear and let ℋ(U) be a σ -invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on U. It is shown that the space ℋ_ϕ (V):={f ∈Hol(V): J_ϕ· f ∘ϕ∈ℋ(U)} endowed with the norm ‖ f‖ _ϕ :=‖ J_ϕ· f ∘ϕ‖ _ℋ(U) is a reproducing kernel Hilbert space and the linear mapping _ϕ defined by _ϕ (f) = J_ϕ· f ∘ϕ , f ∈Hol(V), is a unitary from ℋ_ϕ (V) onto {f ∈ℋ(U): f = -f ∘σ}. Moreover, a neat formula for the reproducing kernel κ _ϕ of ℋ_ϕ (V) in terms of the reproducing kernel of ℋ(U) is given. The above scheme is applicable to symmetrized bidisc, tetrablock, d-dimensional fat Hartogs triangle and d-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann’s inequality for contractive tuples naturally associated with these domains.

We introduce and study Dirichlet-type spaces $\mathcal D(\mu_1, \mu_2)$ of the unit bidisc $\mathbb D^2,$ where $\mu_1, \mu_2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(\mu_1, \mu_2)$ and the complex polynomials are dense in $\mathcal D(\mu_1, \mu_2).$ Further, we obtain the division property and solve Gleason's problem for $\mathcal D(\mu_1, \mu_2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr M_z$ of the multiplication operators $\mathscr M_{z_1},$ $\mathscr M_{z_2}$ on $\mathcal D(\mu_1, \mu_2)$ defines a cyclic toral $2$-isometry and $\mathscr M^*_z$ belongs to the Cowen-Douglas class ${\bf B}_1(\mathbb D^2_r)$ for some $r >0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter's representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair $T$ with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(\mu_1, \mu_2)$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

We discuss the problem of classifying polynomials $p : \mathbb R^2_+ \rightarrow (0, \infty)$ for which $\frac{1}{p}=\{\frac{1}{p(m, n)}\}_{m, n \geq 0}$ is joint completely monotone, where $p$ is a linear polynomial in $y.$ We show that if $p(x, y)=a+b x+c y+d xy$ with $a > 0$ and $b, c, d \geq 0,$ then $\frac{1}{p}$ is joint completely monotone if and only if $a d - b c \leq 0.$ We also present an application to the Cauchy dual subnormality problem for toral $3$-isometric weighted $2$-shifts.

Let T = (T1, T2) be a pair of commuting operators such that the Taylor spectrum sigma(T) of T is contained in the closed unit bidisc D2 and the left spectrum of either of T1 or T2 is contained in the unit circle partial differential D. If the core operator of T is negative, then we show that sigma(T) is either contained in partial differential D2 or equal to D2. This fact applies in particular to a commuting pair of m-isometries with negative core operator. Our method of proof relies on a strictly 2 -variable fact about the topological boundary of the Taylor spectrum. We also present several applications of this dichotomy to the multivariate Fredholm theory. (c) 2024 Elsevier Inc. All rights reserved.

We study B-operators (Brownian-type operators), which are upper triangular 2 x 2 block matrix operators with entries satisfying some algebraic constraints. We establish a lifting theorem stating that any B-subnormal operator, i.e., a B-operator with subnormal (2,2) entry, lifts to a B-normal operator, i.e., a B-operator with normal (2,2) entry, where lifting is understood in the sense of extending entries of the block matrices representing the operators in question. The spectral inclusion and the filling-in-holes theorems are obtained for such operators.

We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$ -isometries with $m > 2$ . We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger–Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $\mathcal {A}_1$ , in the chain $\mathcal {A}_0 \subseteq \mathcal {A}_1 \subseteq \ldots \subseteq \mathcal {A}_{\infty }$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $\mathcal {A}_k$ is finite or not. In particular, a Berger–Shaw-type theorem fails to be true for members of $\mathcal {A}_{\infty }$ . This discrepancy is better revealed in the context of $C^*$ - and $W^*$ -algebras.

The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Krakow school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics.

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2x2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strongand the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed.

For a matrix a=(a_m, n)_m, n=1^∞ with scalar entries, the Dirichlet series kernel κ _a is the double Dirichlet series κ _a(s, u) = ∑ _m, n =1^∞ a_m, nm^-s n^-u in the variables s and u, which is regularly convergent on some right half-plane ℍ_ρ . The analytic symbols A_n, a = ∑ _m=1^∞a_m, nm^-s, n ⩾ 1 play a central role in the study of the reproducing kernel Hilbert space ℋ_a associated with the positive semi-definite kernel κ _a. In particular, they form a total subset of ℋ_a and provide the formula ∑ _n=1^∞⟨f, A_n, a⟩ n^-s, s ∈ℍ_ρ , for f ∈ℋ_a. We combine the basic theory of Dirichlet series kernels with the Gelfond-Schneider theorem (Hilbert’s seventh problem) to show that any quasi-invariant Dirichlet series kernel κ _a(s, u) factors as f(s)f(u) for some Dirichlet series f on ℍ_ρ . In particular, there is no quasi-invariant Dirichlet series kernel κ _a if the dimension of ℋ_a is bigger than one. This is in strict contrast with the case of the unit disc, where non-factorable quasi-invariant kernels exist in abundance. We further discuss the Dirichlet series kernels κ _a invariant under the group 𝒯 of translation automorphisms of ℍ_ρ and construct a family of densely defined 𝒯 -homogeneous operators in ℋ_a, whose adjoints are defined only at the zero vector.

We consider the family 𝒫 of n-tuples P consisting of polynomials P_1, … , P_n with nonnegative coefficients, which satisfy ∂ _i P_j(0) = δ _i, j, i, j=1, … , n. With any such P, we associate a Reinhardt domain ^n__P that we call the generalized Hartogs triangle. We are particularly interested in the choices P_a = (P_1, a, … , P_n, a), a ⩾ 0, where P_j, a(z) = z_j + a ∏ _k=1^n z_k, j=1, … , n. The generalized Hartogs triangle associated with P_a is given by: ^n_a= {z ∈ℂ×ℂ^n-1_*: |z_j|^2< |z_j+1|^2(1-a|z_1|^2), j=1, … , n-1, |z_n|^2 + a|z_1|^2 < 1}. The domain ^2__0 is the Hartogs triangle. Unlike most domains relevant to the multi-variable operator theory, the domain ^n__P, n ⩾ 2, is never polynomially convex. However, ^n__P is always holomorphically convex. With any P ∈𝒫 and m ∈ℕ^n, we associate a positive semi-definite kernel 𝒦__P, m on ^n__P. This, combined with the Moore’s theorem, yields a reproducing kernel Hilbert space ℋ^2_m( ^n__P) of holomorphic functions on ^n__P. We study the space ℋ^2_m( ^n__P) and the multiplication n-tuple ℳ_z acting on ℋ^2_m( ^n__P). It turns out that ℳ_z is never rationally cyclic, but ℋ^2_m( ^n__P) admits an orthonormal basis consisting of rational functions on ^n__P. Although the dimension of the joint kernel of ℳ^*_z-λ is constant of value 1 for every λ∈ ^n__P , it has jump discontinuity at the serious singularity 0 of the boundary of ^n__P with the value equal to ∞ . We capitalize on the notion of joint subnormality to define a Hardy space ℋ^2( ^n__0) on the n-dimensional Hartogs triangle ^n__0. This in turn gives an analog of the von Neumann’s inequality for ^n__0.

In this paper, we study Brownian-type operators, which are upper triangular $2\times 2$ block matrix operators with entries satisfying some algebraic constraints. We establish a lifting theorem stating that any Brownian-type operator with subnormal $(2,2)$ entry lifts to a Brownian-type operator with normal $(2,2)$ entry, where lifting is understood in the sense of extending entries of the block matrices representing the operators in question. The spectral inclusion and the filling in holes theorems are obtained for such operators.

Let (S1,S2) be a bi-isometry, that is, a pair of commuting isometries S1 and S2 on a complex Hilbert space H. By the von Neumann-Wold decomposition, the hyper-range H∞(S1):=∩n=0∞S1nH of S1 reduces S1 to a unitary operator. Although H∞(S1) is an invariant subspace for S2, in general, H∞(S1) is not a reducing subspace for S2. We show that H∞(S1) reduces S2 to an isometry if and only if the subspaces S2(ker⁡S1⁎) and H∞(S1) of H are orthogonal. Further, we describe all bi-isometries (S1,S2) satisfying the orthogonality condition mentioned above.

We discuss the question of the analyticity of a rank one perturbation of an analytic operator. If ℳ_z is the bounded operator of multiplication by z on a functional Hilbert space ℋ_κ and f ∈ℋ with f(0)=0, then ℳ_z + f ⊗ 1 is always analytic. If f(0) 0, then the analyticity of ℳ_z + f ⊗ 1 is characterized in terms of the membership to ℋ_κ of the formal power series obtained by multiplying f ( z ) by 1/f(0)-z. As an application, we discuss the problem of the invariance of the left spectrum under rank one perturbation. In particular, we show that the left spectrum σ _l(T + f ⊗ g) of the rank one perturbation T + f ⊗ g, g ∈ (T^*), of a cyclic analytic left invertible bounded linear operator T coincides with the left spectrum of T except the point ⟨f, g⟩ . In general, the point ⟨f, g⟩ may or may not belong to σ _l(T + f ⊗ g). However, if it belongs to σ _l(T + f ⊗ g) \{0}, then it is a simple eigenvalue of T + f ⊗ g .

A rooted directed tree $${\mathscr {T}}=(V, E)$$ with root $${\textsf{root}}$$ can be extended to a directed graph $$\mathscr {T}_\infty =(V_\infty , E_\infty )$$ by adding a vertex $$\infty $$ to V and declaring each vertex in V as a parent of $$\infty $$ . One may associate with the extended directed tree $${\mathscr {T}}_{\infty }$$ a family of semigroup structures $$\sqcup _{\mathfrak b}$$ with extreme ends being induced by the join operation $$\sqcup $$ and the meet operation $$\sqcap $$ from lattice theory (corresponding to $$\mathfrak b={\textsf{root}}$$ and $$\mathfrak b= \infty $$ respectively). Each semigroup structure among these leads to a family of densely defined linear operators $$W^{(\mathfrak b)}_{\varvec{\lambda }_{u}}$$ acting on $$\ell ^2(V),$$ which we refer to as weighted join operators at a given base point $$\mathfrak b \in V_{\infty }$$ with prescribed vertex $$u \in V$$ . The extreme ends of this family are weighted join operators $$W^{(\mathfrak {{\textsf{root}}})}_{\varvec{\lambda }_{u}}$$ and weighted meet operators $$W^{(\mathfrak \infty )}_{\varvec{\lambda }_{u}}$$ . In this paper, we systematically study the weighted join operators on rooted directed trees. We also present a more involved counterpart of weighted join operators $$W^{(\mathfrak b)}_{\varvec{\lambda }_{u}}$$ on rootless directed trees $${\mathscr {T}}$$ . In the rooted case, these operators are either finite rank operators, diagonal operators or rank one perturbations of diagonal operators. In the rootless case, these operators are either possibly infinite rank operators, diagonal operators or (possibly unbounded) rank one perturbations of diagonal operators. In both cases, the class of weighted join operators overlaps with the well-studied classes of complex Jordan operators and n-symmetric operators. An important half of this paper is devoted to the study of rank one extensions $$W_{f, g}$$ of weighted join operators $$W^{(\mathfrak b)}_{\varvec{\lambda }_{u}}$$ on rooted directed trees, where $$f \in \ell ^2(V)$$ and $$g: V \rightarrow {\mathbb {C}}$$ is unspecified. Unlike weighted join operators, these operators are not necessarily closed. We provide a couple of compatibility conditions involving the weight system $$\varvec{\lambda }_u$$ and g to ensure closedness of $$W_{f, g}$$ . These compatibility conditions are intimately related to whether or not an associated discrete Hilbert transform is well-defined. We discuss the role of the Gelfand-triplet in the realization of the Hilbert space adjoint of $$W_{f, g}$$ . Further, we describe various spectral parts of $$W_{f, g}$$ in terms of the weight system and the tree data. We also provide sufficient conditions for $$W_{f, g}$$ to be a sectorial operator (resp. an infinitesimal generator of a quasi-bounded strongly continuous semigroup). In case $${\mathscr {T}}$$ is leafless, we characterize rank one extensions $$W_{f, g}$$ , which admit compact resolvent. Motivated by the above graph-model, we also take a brief look into the general theory of rank one non-selfadjoint perturbations.

The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a 2-isometry is subnormal. In this paper, we address this problem for cyclic 2-isometries. In view of some recent developments in operator theory on function spaces (see [4, 22]), one may recast CDSP as the problem of subnormality of the Cauchy dual M ′ z of the multiplication operator Mz acting on a de BrangesRovnyak space H(B), where B is a vector-valued rational function. The main result of this paper characterizes the subnormality of M ′ z on H(B) provided B is a vector-valued rational function with simple poles. As an application, we provide affirmative solution to CDSP for the Dirichlettype spaces D(μ) associated with measures μ supported on two antipodal points of the unit circle. 1. Cauchy dual subnormality problem for 2-isometries The Cauchy dual subnormality problem (for short, CDSP) for 2-isometries can be seen as the manifestation of the rich interplay between positive definite and negative definite functions on abelian semigroups. Indeed, CDSP can be considered as the non-commutative variant of the fact from the harmonic analysis on semigroups that the reciprocal of a Bernstein function f : [0,∞) → (0,∞) is completely monotone (see [31, Theorem 3.6]). This fact turns out to be somewhat equivalent to the solution of CDSP for completely hyperexpansive weighted shifts (see [7, Proposition 6] for a generalization). Another early result towards the solution of CDSP asserts that the Cauchy dual of any concave operator is a hyponormal contraction (see [33, Equation (26)]). Later this fact was generalized in [13, Theorem 3.1] by deducing power hyponormality of the Cauchy dual of any concave operator. Around the same time CDSP was settled affirmatively for ∆T -regular 2-isometries in [8, Theorem 3.4] and for 2-isometric operator-valued weighted shifts in [5, Theorems 2.5 and 3.3] (see also [14, Corollary 6.2] for the solution for yet another subclass of 2-isometries). Further, it was shown in [5, Examples 6.6 and 7.10] that there exist 2-isometric weighted shifts on directed trees (that include adjacency operators) whose Cauchy dual is not necessarily subnormal. Recently, a class of cyclic 2-isometric composition operators without subnormal Cauchy dual has been exhibited in [6, Theorem 4.4]. 2000 Mathematics Subject Classification. Primary 47B32, 47B38; Secondary 44A60, 31C25.

For a commuting pair T of bounded linear operators T-1 and T-2 on a Hilbert space H, let D-T = T-2*T-2 - T-1*T-1. If T-2*DTT2 <= D-T and the Taylor spectrum of T is contained in the Hartogs triangle Delta(H), then for any bounded holomorphic function phi on Delta(H), parallel to phi(T)parallel to <= parallel to phi parallel to(infinity). We deduce this fact from an analogue of von Neumann's inequality for bounded domains in C-d. The proof of the latter closely follows the model theory approach as developed in [1].