Complex symmetry of invertible composition operators

In this paper, we study 2-complex symmetric composition operators with the conjugation J, defined by Jf(z)=(f(z¯))¯, on the Hardy space H2. More precisely, we obtain the necessary and sufficient condition for the composition operator Cϕ to be 2-complex symmetric with J when ϕ is an automorphism of D. We also characterize 2-complex symmetric with J when ϕ is a linear fractional self-map of D.

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

The composition operator induced by a holomorphic self-map of the unit disc is compact on L 1 {L^1} of the unit circle if and only if it is compact on the Hardy space H 2 {H^2} of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on L 1 {L^1} is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on H 2 {H^2} .

Approximation numbers of weighted composition operators

In this article, we introduce a new class of conjugations in the scalar-valued Hardy space and provide a characterization of a complex symmetric Toeplitz operator with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator on the vector-valued Hardy space with respect to certain conjugations introduced in [Câmara MC, Kliś-Garlicka K, Ptak M. Complex symmetric completions of partial operator matrices. Linear and Multilinear Algebra. 2019; DOI: 10.1080/03081087.2019.1631246], [Kang D, Ko E, Lee JE. Remarks on complex symmetric Toeplitz operators. Linear Multilinear Algebra. 2020; DOI: 10.1080/03081087.2020.1842847], [Ko E, Lee JE. Remark on complex symmetric operator matrices. Linear Multilinear Algebra. 2019;67(6):1198–1216].

ABSTRACTLet μ be a normal function on . Suppose ϕ is a holomorphic self-map on the unit ball B of . In this paper, the authors characterize the conditions that the composition operator is bounded or compact on the normal weight Zygmund space when n>1. At the same time, the authors give the sufficient and necessary conditions for some special cases.

In this paper, we investigate the conditions under which the Toeplitz composition operator on the Hardy space H2 becomes complex symmetric with respect to a certain conjugation. We also study various normality conditions for the Toeplitz composition operator on H2.

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H1 whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula A≥E, B(x)≥0, and the operator B(x) is bounded for all x. An operator L0 is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) eH1 according to the formula: L0y=−y″ + Q(x)y (−∞<x<∞). It is proved that the closure of the operator L0 is a self-adjoint operator in H1 under the given assumptions.

We investigate the composition operators on the weighted Hardy spaces H2(β). For any bounded weight sequence β, we give necessary conditions for those operators to be isometric. The sufficiency of those conditions is well‐known for the classical space H2. In the case where β is non‐decreasing or non‐increasing, their sufficiency holds only for very few weighted spaces. We find out such spaces by characterizing the isometric monomial composition operators, first for a general β, then for any β as before. With no restriction on β, we provide a complete description of all isometric composition operators. We also prove that the unitary monomial ones are the same as those acting on H2. Such a fact extends to general symbols in the case where β is bounded (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Introduction. It has been known for some time that on a complex infinite-dimensional Hilbert space there exist invertible operators without square roots, indeed without roots of any order, which therefore do not belong to the range of the exponential function. A first class of examples of such operators was described by Halmos, Lumer, and Schaffer [3]: the space considered was the separable Hilbert space of all complex-valued functions defined, analytic, and squaresummable on a domain D of the complex plane (with the L2-norm); the operator was the analytic position operator A defined by (A+5) (z) =zcf(z), zCD. It was shown that A is invertible and lacks a square root (indeed, a root of any order) if and only if D surrounds the origin but does not contain it. Halmos and Lumer [2] used the concept of multiplicity to show that the analytic position operator for such D is an interior point (in the norm topology for operators) of the set of invertible operators without roots. Recently, Deckard and Pearcy [1] described another class of invertible square-root-less operators on a separable Hilbert space; the spectral properties of these operators are quite different from those of the operators discussed in [3] and [2]. We shall not be concerned with this class of examples in this paper. Root-less operators are beginning to be used in various contexts: we mention, in particular, the disproof by Massera and Schaffer [5, p. 92], of an extension to Hilbert space of Floquet's Theorem on periodic differential equations. It appears that more information about such operators and more flexibility in their choice is desirable. To take the analytic position operator, for instance: there is of course some latitude in the choice of the domain D, but the description of the operator in terms of an orthonormal basis is awkward even in the simplest case, when D is an annulus centred at the origin, and quite unmanageable in other cases; besides, this description can not be perturbed or altered in any prescribed way as the application in hand might require. We remark, however, that the original definition has the advantage of making almost obvious the spectral structure that ensures the lack of roots.

We characterise the complex measures μ on the open unit disk D such that for all f in the Hardy space H2. The characterisation involves Carleson measures, the duality between H1 and BMOA, and Hankel operators.

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 $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C