Normal and quasinormal composition operators

Let Cφ be a composition operator on L2(λ), where λ is a σ-finite measure on a set X. Conditions under which Cφ is invertible, unitary, and normal are investigated in this paper.

Few, if any, of the properties enjoyed by ergodic measure-preserving transformations defined on a finite measure space generalize in a natural way to those defined on an infinite measure space. Concrete examples of ergodic transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature, see [1]. It is not difficult to see that ergodic transformations never admit wandering sets of positive measure. In [2 ] it was shown that a basic difference exists between ergodic transformations which preserve a finite measure and those which preserve an infinite measure; namely, an ergodic measure-preserving transformation defined on an infinite measure space always admits weakly wandering sets of positive measure (Theorem 2 of [2]). Unlike wandering sets, it is not true in general that the union of two weakly wandering sets is again a weakly wandering set even if we require that a class of mutually disjoint images of one weakly wandering set does not intersect a class of mutually disjoint images of the other. One may ask then if there are any ergodic measure-preserving transformations defined on an infinite measure space which admit only weakly wandering sets of finite measure. In this paper we show that this is not the case. We prove that there always exist weakly wandering sets of infinite measure for any ergodic measure-preserving transformation defined on an infinite measure space (Theorem 3). In [2] the existence of a weakly wandering set of positive measure for an ergodic measure-preserving transformation defined on an infinite measure space was discovered while studying the necessary and sufficient conditions for the existence of a finite, invariant, and equivalent measure for a given measurable and nonsingular transformation. In this paper we construct the weakly wandering sets in a different way. Using the pointwise ergodic theorem we prove a simple and yet a useful fact about ergodic measure-preserving transformations defined on an infinite measure space (Theorem 2). It states that given two sets A and B both of finite measure, it is possible to find an image of A under some power of the transformation T which has small intersection with the set B. This fact is basic in proving Lemma 1 which shows the existence of a

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

Partially normal composition operators relevant to weighted directed trees

Bayes spaces are vector spaces of sigma-additive positive measures. Proportional measures are considered equivalent and can be represented by densities with respect to a fixed dominating measure. The addition in these spaces is perturbation. It corresponds to Bayes theorem, which appears as a linear operation. Bayes spaces, with continuous dominating measures, contain finite and infinite measures. Finite measures are equivalent to probability measures. Infinite measures include what in Bayesian statistics are called improper priors and non-integrable likelihood functions, justifying the use of such improper densities in Bayes theorem. Many concepts of probability theory can be handled in a natural way in the context of Bayes spaces. Particularly, an exponential family of probability densities appears as a cone contained in an affine subspace of the Bayes space. The framework of Bayes spaces allows an easy handling of exponential families and their extensions to improper distributions. Furthermore, the vector space structure of Bayes spaces allows the definition of derivatives of densities. In Bayesian statistics, these derivatives are a new tool to examine sensitivity of posterior distributions with respect to both observed data and prior changes.

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_{\alpha}^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $\varphi(z)=$ $z+\psi(z)$, where $\psi\in$ $H^{\infty}(\mathbb{H})$ and $\Im(\psi(z)) > \epsilon > 0$. We especially examine the case where $\psi$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

Mapping properties that preserve convergence in measure on finite measure spaces

We show that an ergodic measure preserving action Γ ↷ ( X , μ ) \Gamma \curvearrowright (X,\mu ) of a discrete group Γ \Gamma on a σ \sigma -finite measure space ( X , μ ) (X,\mu ) satisfies the local spectral gap property introduced in Invent. Math. 208 (2017), 715–802, if and only if it is strongly ergodic. In fact, we prove a more general local spectral gap criterion in arbitrary von Neumann algebras. Using this criterion, we also obtain a short proof of Connes’ spectral gap theorem for full I I 1 \mathrm {II}_1 factors as well as its recent generalization to full type I I I \mathrm {III} factors.

Let T T be a bijective nonsingular transformation on a finite measure space. We shall first construct a σ \sigma -finite and finite invariant measure by a unified method which is valid for both cases. Secondly we shall give another construction of a finite invariant measure. We shall also give a new necessary and sufficient condition of a unified form for the existence of σ \sigma -finite and finite invariant measures. Further, we shall discuss in detail ergodic transformations.

If the Lp spaces of two measure spaces are “the same”, to what extent can we identify the underlying measure spaces? This question has been partially answered by Schneider [7] (extending results of Forelli [2]). He proves that a linear isometry between the Lv spaces of two finite measure spaces is in fact an (isometric) homomorphism between the corresponding L∞ spaces, if it preserves the identity.

Let X \mathfrak {X} be a Banach space and ( Ω , Σ , μ ) (\Omega ,\Sigma ,\mu ) be a finite measure space. A strongly measurable f : Ω → X f:\Omega \to \mathfrak {X} is Pettis integrable if and only if there exists a Young’s function Φ \Phi with lim t → ∞ Φ ( t ) / t = ∞ {\lim _{t \to \infty }}\Phi (t)/t = \infty such that x ∗ f ∈ L Φ ( μ ) {x^ \ast }f \in {L^\Phi }(\mu ) for all x ∗ ∈ X ∗ {x^ \ast } \in {\mathfrak {X}^ \ast } .

Introduction.Let T be a conservative positive contraction on the Lx space of a finite measure space (X, &, ft).A theorem of Chacon [5], [2] shows that T defines a sub CT-field J of J^ consisting of invariant subsets of X.The ratio ergodic limits are measurable with respect to J [5], [2] and the class of these limits contains Lm(X, J, ft), which can be considered as the invariant functions of the adjoint transformation [2].The main purpose of the present paper is to show that any positive contraction on LX(X, 3F, ft) behaves, asymptotically, like a conservative transformation (Theorems 3 and 4) and that the invariant functions of the adjoint transformation can be approximated by the ratio ergodic limits.Intuitively, a ratio ergodic limit corresponds to the result of an averaging process of different values of a function.It is then natural to consider these limits as functions that are smooth with respect to the asymptotic behaviour of the transformation.This leads (Theorem 6) to a Martin-Doob type representation [12], [8] of invariant functions as the L functions of a compact Hausdorff space J with a Baire measure.The topology on Ji is just strong enough to make the ergodic limits to correspond to continuous functions.As an example we consider a transformation of Feller [10] and show that for this case the above representation is identical with the Poisson representation of harmonic functions in the unit disk.We also consider the possibility of joining X and J, convergence of measures to i' in IuJ (Theorem 7), and a relation (Lemma 9) between the Feller and Martin-Doob type representations, corresponding to a result of Feldman [9].2. Preliminaries.Let (X, F,p) be a finite measure space and let Lp = Lv(X,!F,p), la/?^oo be the usual Banach spaces, and Lp denote the positive cone of Lp.Let T: Lx -* Lx be a positive linear contraction and U: LOT -> Lx be its dual.For oei" define Ta:Lx^ Lx as Taf=af+ T(l-a)fife Lx, and let Ua be its dual.If xe is the characteristic function of E e F we write TE and UE instead of TXE and UXs.The following partial ordering of Lx is similar to that of Bishop and deLeeuw given in [3].Definition 1.For fige Lx, /-<g if and only if there exist an integer 1 and ax,.. .,aneLxsuch that O^a,^ 1 for /=1,..., and g=Tan- -Taif.This relation is reflexive and transitive and f<g implies ||/|i^ ||g||i.Also, an

Multicyclicity of unbounded normal operators and polynomial approximation in C

A bounded linear operator between Banach spaces is called a Dieudonné operator (=weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $X$ and

Linear operators associated with p-Dunford, p-Pettis and p-Bochner integrable functions with values in a Banach space