J. E. Pascoe
We give an entire free holomorphic function $f$ which is unbounded on the row ball. That is, we give a holomorphic free noncommutative function which is continuous in the free topology developed by Agler and McCarthy but is unbounded on the set of row contractions.
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We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
The theory of characteristic functions for row contractions is used to determine the group $Aut(B(H)^n_1)$ of all free holomorphic automorphisms of the unit ball of $B(H)^n$. We show that the noncommutative Poisson transform commutes with the action of the automorphism group $Aut(B(H)^n_1)$. This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra $C^*(S_1,..., S_n)$, which leave invariant the noncommutative disc algebra $\cA_n$. This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals n…
In this paper, we introduce a unitary invariant $Γ$ defined on the unit ball of $B(H)^n$ in terms of the characteristic function, the noncommutative Poisson kernel, and the defect operator associated with a row contraction. We show that $Γ$ detects the pure row isometries and completely classify them up to a unitary equivalence. We also show that $Γ$ detects the pure row contractions with polynomial characteristic functions and completely non-coisometric row contractions. In particular, we show that any completely non-coisometric row contraction with constant characteristic function is homogen…
A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call {\sl pseudo-contractivity}, is introduced by means of the Abel averages of such maps. We show that the studied maps are dissipative in the spirit of the classical Lumer-Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.
In this article we prove a general result which in particular suggests that, on a simply connected domain in C, all the derivatives and anti-derivatives of the generic holomorphic function are unbounded. A similar result holds for the operator of partial sums of the Taylor expansion with center z at 0, seen as functions of the center z. We also discuss a universality result of these operators.
This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. The extend the Hille--Phillips calculus for (negative) generators $A$ of certain bounded $C_0$-semigroups, in particular for bounded semigroups on Hilbert spaces and bounded holomorphic semigroups on Banach spaces. They include functions outside the Hille-Phillips class, and they generally give sharper bounds for the norms of the resulting operators $f(A)$. The calculi are mostly based on appropriate reproducing formulas for the relevant classes of functions, and…
A Survey on Invariant Spaces of Holomorphic Functions on Symmetric Domains
本文综述了对称域上全纯函数不变空间的研究，涵盖加权伯格曼空间、Hardy空间等，并重点探讨了Hilbert和半Hilbert空间，同时讨论了最小和最大空间。
We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic function is not - even locally - in Bergman spaces of higher order. Finally, in certain domains we consider the space of holomorphic functions whose derivatives up to some order extend continuously to the closure of the domain endowed with its natural topology. Generically, every function in this space is proven to be non - extendable, although bounded.
Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces
本文研究了有限秩德布兰斯-罗文纳克空间H(B)的函数理论方面，通过推广萨拉斯逊的工作，将有限秩H(B)-空间表征为托利齐亚算子T_φ*的伴随算子的域，并推导了H(B)-空间中函数的范数公式，同时应用该结果扩展了萨拉斯逊对经典德布兰斯-罗文纳克空间的判据。
Holomorphic Functional Calculus approach to the Characteristic Function of Quantum Observables
本文提出用柯西积分公式与邓福德全纯泛函微积分计算量子随机变量的真空特征函数，并详细分析了由位置和动量算符定义的若干量子可观测量的情形。
Doubling measures and Poincaré inequalities for sphericalizations of metric spaces, with applications to $p$-harmonic functions in unbounded domains
本文将度量空间球化的适用条件从Ahlfors正则放宽为加倍测度，证明其保持p-调和函数与庞加莱不等式，进而得到无界区域p-调和函数的相关新结果。
In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obt…
We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not…
Let $A(t)$ be a holomorphic family of self-adjoint operators of type (B) on a complex Hilbert space $\mathcal{H}$. Kato-Rellich perturbation theory says that isolated eigenvalues of $A(t)$ will be analytic functions of $t$ as long as they remain below the minimum of the essential spectrum of $A(t)$. At a threshold value $t_0$ where one of these eigenvalue functions hits the essential spectrum, the corresponding point in the essential spectrum might or might not be an eigenvalue of $A(t_0)$. Our results generalize a theorem of Simon to give a sufficient condition for the minimum of the essentia…
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