Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit ball. In this paper, we discuss a class of maps in $\mathbb {C}^N$ that generalize linear fractional maps. We then proceed to determine precisely when such a map is a self-map of the unit ball. In particular, we take a novel approach, obtaining numerous new results about this class of maps along the way.

We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $\operatorname {Lip}(X,E)$, where E is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $\operatorname {Lip}(X,\mathcal {A})$, where $\mathcal {A}$ is a unital factor $C^{*}$-algebra. These results improve previous results stated by the author.

We study the relationship between weak* Dunford–Pettis and weakly (resp. M-weakly, order weakly, almost M-weakly, and almost L-weakly) operators on Banach lattices. The following is one of the major results dealing with this matter: If E and F are Banach lattices such that F is Dedekind $\sigma $-complete, then each positive weak* Dunford–Pettis operator $T:E\rightarrow F$ is weakly compact if and only if one of the following assertions is valid: (a) the norms of $E^{\prime }$ and F are order continuous; (b) E is reflexive; and (c) F is reflexive.

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$. We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu $-almost surely. We apply this result to the random logistic map $X_{n+1} = \omega _n X_n (1-X_n)$ absorbed at ${\mathbb R} \setminus [0,1],$ where $\omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

In this article, the question of whether the Löwner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo–Ando mean is studied. The question was affirmatively answered for certain classes of Kubo–Ando means, yet the general case was left as an open problem. We here give a complete answer to this question, by showing that the norm of every symmetric Kubo–Ando mean is order-determining, i.e., if $A,B\in \mathcal B(H)^{++}$ satisfy $\Vert A\sigma X\Vert \le \Vert B\sigma X\Vert $ for every $X\in \mathcal {A}^{{++}}$, where $\mathcal A$ is the C*-subalgebra generated by $B-A$ and I, then $A\le B$.

First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{**}$.

For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$, we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $, respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$, written as $H(\varphi )$ and $H(\overline \varphi )$ and equipped with appropriate inner products, are called sub-Bergman spaces.

We prove the following three results in the paper: for $-1<\alpha \le 0$, the space $H(\varphi )$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$, we have $H(\varphi )=H(\overline \varphi )=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$, we have $D^2_\varphi (A^2_\alpha )= D^2_{\overline \varphi }(A^2_\alpha )=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible.

We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.

Suppose $2< p<\infty$ and $\varphi$ is a holomorphic self-map of the open unit disk $\mathbb {D}$. We show the following assertions:

1. (1) If $\varphi$ has bounded valence and0.1

   \begin{equation} \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2}\frac{\mathrm{d} A(z)}{(1-|z|^2)^2}<\infty, \end{equation}
   then $C_{\varphi }$ is in the Schatten $p$-class of the Hardy space $H^2$.

2. (2) There exists a holomorphic self-map $\varphi$ (which is, of course, not of bounded valence) such that the inequality (0.1) holds and $C_{\varphi }: H^2\to H^2$ does not belong to the Schatten $p$-class.

On all Bergman–Besov Hilbert spaces on the unit disk, we find self-adjoint weighted shift operators that are differential operators of half-order whose commutators are the identity, thereby obtaining uncertainty relations in these spaces. We also obtain joint average uncertainty relations for pairs of commuting tuples of operators on the same spaces defined on the unit ball. We further identify functions that yield equality in some uncertainty inequalities.

In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$, $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ($1\le i\le m$), which are a class of non-trivial subspaces of ${\rm BMO}$, the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$. Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$.

In this paper, we introduce the spherical polar decomposition of the linear pencil of an ordered pair $\mathbf {T}=(T_{1},T_{2})$ and investigate nontrivial invariant subspaces between the generalized spherical Aluthge transform of the linear pencil of $\mathbf {T}$ and the linear pencil of the original pair $\mathbf {T}$ of bounded operators with dense ranges.

In this article, we give a representation of bounded complex linear operators that preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is, moreover, positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group antihomomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realized by the idempotent preserving operators.

We characterize the membership in the Schatten ideals $\mathcal {S}_p$, $0<p<\infty $, of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$ -invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$-valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$, $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$, where $\{\vartheta_n\}_{n \ge 1}$ are independent and identically distributed random variables with common probability distribution $\mathfrak{p}$, $F(\cdot,\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\{\vartheta_n\}_{n \ge 1}$. Under parametric perturbation of both F and $\mathfrak{p}$, we are interested in the robustness of the V-geometrical ergodicity property of $\{X_n\}_{n\in{\mathbb{N}}}$, of its invariant probability measure, and finally of the probability distribution of $X_n$. Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

This paper mainly considers the problem of generalizing a certain class of analytic functions by means of a class of difference operators. We consider some relations between starlike or convex functions and functions belonging to such classes. Some other useful properties of these classes are also considered.

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.

Given a weighted shift T of multiplicity two, we study the set $\sqrt {T}$ of all square roots of T. We determine necessary and sufficient conditions on the weight sequence so that this set is non-empty. We show that when such conditions are satisfied, $\sqrt {T}$ contains a certain special class of operators. We also obtain a complete description of all operators in $\sqrt {T}$.