M‐quasi‐hyponormal composition operators

Let \Omega be an open subset of \mathbb{R}^n . Let f be a Borel measurable function from \mathbb{R} to \mathbb{R} . We prove necessary and sufficient conditions on f in order that the composite function T_f[g]=f\circ g belongs to a generalized Morrey space \mathcal{M}_p^w(\Omega) whenever g belongs to \mathcal{M}_p^w(\Omega) . Then we prove necessary conditions and sufficient conditions on f in order that the composition operator T_f[{\cdot}] be continuous, uniformly continuous, Hölder continuous and Lipschitz continuous in \mathcal{M}_p^w(\Omega) . We also consider its ‘vanishing’ generalized Morrey subspace \mathcal{M}_p^{w,0}(\Omega) and prove the related results for the composition operator as operator acting from \mathcal{M}_p^{w,0}(\Omega) to \mathcal{M}_p^w(\Omega) and also between the spaces \mathcal{M}_p^{w,0}(\Omega) . For the uniform, Hölder and Lipschitz continuity, we also have conditions that are both necessary and sufficient. We also have both necessary and sufficient conditions for the continuity under certain additional natural assumptions. We also consider the most commonly used Morrey classes that are related to power-type weights in the context of a discussion of some of the conditions that we impose on the weights.

In this paper, we study quasicompact and Riesz composition operators on Banach spaces of Lipschitz–Holder functions on pointed metric spaces. For a composition operator T on these spaces, we give an upper bound for $$r_e(T)$$ , the essential spectral radius of T, and establish a formula for $$r_e(T)$$ whenever metric spaces are compact. We also give some necessary and some sufficient conditions that a composition operator T on these spaces to be quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz composition operator on these spaces.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

(Communicated by G. Sinnamon) Abstract. Suppose ϕ is an analytic self-map of open unit disk D and w is an analytic func- tion on D. Then a weighted composition operator induced by ϕ with weight w is given by (Ww,ϕ f)(z )= w(z)f(ϕ(z)) ,f orz ∈ D and f analytic on D.W efind a sufficient condition un- der which two composition operators lie in the same path component of C(H 2 ) ,a nd wefind as ufficient condition for the difference of such operators to be compact on H 2 (D) .T hen we provide another example that answers a question raised by Shapiro and Sundberg (18 )n ega- tively. Moreover, we characterize the Hilbert-Schmidt difference of two composition operators on H2(D).

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.

Let \Omega be a subdomain of \mathbb{C} and let \mu be a positive Borel measure on \Omega . In this paper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operators T_\mu acting on Bergman spaces on \Omega . Let (\lambda _n(T_\mu)) be the decreasing sequence of the eigenvalues of T_\mu , and let \rho be an increasing function such that \rho (n)/n^A is decreasing for some A>0 . We give an explicit necessary and sufficient geometric condition on \mu in order to have \lambda _n(T_\mu)\asymp 1/\rho (n) . As applications, we consider composition operators C_\varphi , acting on some standard analytic spaces on the unit disc \mathbb{D} . First, we give a general criterion ensuring that the singular values of C_\varphi satisfy s_n(C_\varphi ) \asymp 1/\rho(n) . Next, we focus our attention on composition operators with univalent symbols, where we express our general criterion in terms of the harmonic measure of \varphi (\mathbb{D}) . We finally study the case where \partial \varphi (\mathbb{D}) meets the unit circle in one point and give several concrete examples. Our method is based on upper and lower estimates of the trace of h(T_\mu) , where h is a suitable concave or convex function.

We present a non-weak supercyclicity criterion for vectors in infinite dimensional Banach spaces. Also, we give sufficient conditions under which a class of weighted composition operators on a Banach space of analytic functions is not weakly supercyclic. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), is not weakly supercyclic. Moreover, we observe that every composition operator on some Banach space of analytic functions such as the disc algebra or the analytic Lipschitz space is not weakly supercyclic.

In a previous paper, we characterized the Devaney chaos, frequent hypercyclicity and dense distributional chaos of composition operators induced by continuous self-maps on the real line. The present paper further investigates the disjoint hypercyclicity and topological entropy of these operators. It is shown that the composition operator is [Formula: see text]-transitive if and only if it is Cesàro-hypercyclic, if and only if it is supercyclic, if and only if it has the specification property on the whole space. Furthermore, sufficient and necessary conditions for a pair of composition operators to be disjoint hypercyclic (disjoint mixing, respectively) are obtained. Finally, sufficient conditions for the composition operator to admit infinite topological entropy are provided.

Composition operators on Hilbert spaces of entire Dirichlet series

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections onK(X, Y) for Banach spacesX, Y. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result forL(X, Y) thereby proving an analogue of the result from [9] forL p (1 <p ≠ 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

In this paper, we define some new hyperbolic classes of functions, which will be called Bα,log* and Flog*(p,q,s) classes. Then, we introduce D-metrics in these defined hyperbolic classes. Also, these classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, necessary and sufficient conditions are given for a composition operator Cφ to be bounded and compact from Bα,log* to Flog*(p,q,s).

We consider the topological space of all composition operators, acting on certain Hilbert spaces of holomorphic functions on the unit disc, in the uniform operator topology. A sufficient condition is given for the component of a composition operator to be a singleton. A necessary condition is given for one composition operator to lie in the component of another. In addition, we prove analogous results for the component of the image of a composition operator in the Calkin algebra. Finally, we obtain some related results on the essential norm of a linear combination of composition operators.

We find necessary and sufficient conditions on g such that / g (resp.g f) is a function of bounded variation for all/of bounded variation.Let I = [0, 1], and let BV denote the set of functions /: I -> I of bounded variation.BV is not closed under composition (example: / g where/(x) = Vjc; g(0) = 0, g(x) = x2 sin2(l/x) for x ^ 0).Here we characterize those functions g whose right (resp.left) composition operation/-/ g (resp./-*g f) preserves BV.If /: I -> R, define t>(/, a, b) as the variation of / on the subinterval [a, b], and v(f) = t>(/, 0, 1).For a positive integer N, let JN = {X Q I: X can be expressed as a union of A' intervals} (where the intervals may be open or closed at either end and we allow singletons as degenerate closed intervals).Since any interval is a union of two subintervals, we see JN c Jn+v A function/: I-R is said to be of N-bounded variation if f~\[a, b]) G JN for all [a, b] Q R. Let B\(N) be the set of all functions /: I -* I of N-bounded variation, and BV'(N) the set of all bounded functions /: I -> R of N-bounded variation.Lemma 1.Every function in BV'(JV) is of bounded variation.Proof.Suppose h G BV'(N) and \h(x)\ < M for all x G I. We show v(h) < 4M(N + 1).Ift>(A) > 4M(N + 1), then there exists a partition {x0, . . ., x") of I with |A(*,)-/(*,_,)I>4M(JV+1).<-l Some [a', b'] C [-M, M] with a! < b' is covered more than 2(N + 1) times by intervals [/(x,_,), h(x)] or [A(x,),/t(x,_,)].At least N + 1 are of the type [(*,-i), Kx)], say, so N+l [a',b']c n[Hyj),h(zj)] y-i where each (yJtz,) = (jc,-_" x) for some i, and yx <zx < <yN+x <zN+x.Then h{yf) < a' < b' < h(zf) for all/.Take a between a' and b'; take b greater than

In this paper, some sufficient and necessary conditions for the composition operator C? to be 2-normal are investigated when the symbol ? is a linear fractional self-map of D.

In this paper, we study single variable functional equations that involve one unknown function and a finite set of known functions that form a group under the operation of composition. The main theorems give sufficient conditions for the existence and uniqueness of a (local) solution and also stability-type result for the solution. In the proofs, besides the standard methods of classical analysis, some group theoretical tools play a key role.