M. Lilian Lourenço, Daniela M. Vieira
In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E into E, which are not Lorch-analytic.
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We say that a function $f:[0,1]\rightarrow \R$ is \emph{nowhere $L^q$} if, for each nonvoid open subset $U$ of $[0,1]$, the restriction $f|_U$ is not in $L^q(U)$. For a fixed $1 \leq p <\infty$, we will show that the set $$ S_p\doteq {f \in L^p[0,1]: f is nowhere $L^q$, for each p<q \leq \infty}, $$ united with ${0}$, contains an isometric and complemented copy of $\ell_p$. In particular, this improves a result from G. Botelho, V. Fávaro, D. Pellegrino, and J. B. Seoane-Sepúlveda, $L_p[0,1]\setminus \cup_{q>p} L_q[0,1]$ is spaceable for every $p>0$, preprint, 2011., since $S_p$ turns out to be…
Some results about spaceability in function spaces
研究无处正则周期函数集合的空间性，并证明实解析函数在C((0,1))中具有空间性，同时给出由实解析函数构成的闭子空间的完全刻画。
In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup $(\v_t)$ of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Alexandrov-Clark measures techniques. In particular we prove that the Alexandrov-Clark measure of $(\v_t)$ at a boundary regular fixed points is differentiable (in the weak$^\ast$-topology) with respect to $t$.
This paper provides a complete characterization of quasicontractive groups and analytic $C_0$-semigroups on Hardy and Dirichlet space on the unit disc with a prescribed generator of the form $Af=Gf'$. In the analytic case we also give a complete characterization of immediately compact semigroups. When the analyticity fails, we obtain sufficient conditions for compactness and membership in the trace class. Finally, we analyse the case where the unit disc is replaced by the right-half plane, where the results are drastically different.
No. The title question was posed by D. Kalyuzhnyi-Verbovetskyi [1, Problem 1.3]. Let \mathcal{L(H,K)} denote the set of all bounded linear operators between a pair of Hilbert spaces \mathcal{H,K}, and let \mathbb{D}^{n} and \mathbb{T}^{n} denote the open unit polydisk, and the unit n-torus, respectively.
We study the boundedness of Toeplitz operators $T_ψ$ with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball of $\mathbb{R}^n$. Generalizing earlier results for analytic function spaces, we derive a general sufficient condition for the boundedness of $T_ψ$ in terms of suitable averages of its symbol. We also obtain a similar "vanishing" condition for compactness. Finally, we show how these results can be transferred to the setting of the standard weighted Bergman spaces of analytic functions.
We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.
We study weighted $H^\infty$ spaces of analytic functions on the open unit disc in the case of non-doubling weights, which decrease rapidly with respect to the boundary distance. We characterize the solid hulls of such spaces and give quite explicit representations of them in the case of the most natural exponentially decreasing weights.
We study biorthogonal sequences with special properties, such as weak or weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig theorem. This result is applied to embed analytic disks in the fiber over 0 of the spectrum of H^infinity (B), the algebra of bounded analytic functions on the unit ball B of an arbitrary infinite dimensional Banach space. Various other embedding theorems are obtained. For instance, if the Banach space is superreflexive, then the unit ball of a Hilbert space of uncountable dimension can be embedded analytically in the fiber over 0 via an embed…
Let $Ω\subset \mathbb{C}^m$ be an open, connected and bounded set and $\mathcal{A}(Ω)$ be a function algebra of holomorphic functions on $Ω$. In this article we study quotient Hilbert modules obtained from submodules, consisting of functions in $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subsetΩ$ of codimension at least $2$, of a quasi-free Hilbert module, $\mathcal{M}$. Our motive is to investigate unitary invariants of such quotient modules. We completely determine unitary equivalence of aforementioned quotient modules and relate it to ge…
For a rational matrix function $Φ$ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation $\AΦ$ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of $2\times2$ matrix functions. It turns out that ``generically'' $°\AΦ\le\degΦ-2$. We prove that for an arbitrary $2\times2$ rational function $Φ$, $°\AΦ\le2\degΦ-3$ whenever $\degΦ\ge2$. On the other hand, for $k\ge2$, we construct a $2\times2$ matrix function $Φ$, for which $\degΦ=k$, while $°\AΦ=2k-3$. Moreover, we conduct a detailed analysis of the situation when the…
We prove that, for every $α> -1$, the pull-back measure $ϕ({\cal A}_α)$ of the measure $d{\cal A}_α(z) = (α+ 1) (1 - |z|^2)^α\, d{\cal A} (z)$, where ${\cal A}$ is the normalized area measure on the unit disk $\D$, by every analytic self-map $ϕ\colon \D \to \D$ is not only an $(α+ 2)$-Carleson measure, but that the measure of the Carleson windows of size $\eps h$ is controlled by $\eps^{α+ 2}$ times the measure of the corresponding window of size $h$. This means that the property of being an $(α+ 2)$-Carleson measure is true at all infinitesimal scales. We give an application by characterizing…
A characterization of those points in the unit disc which belong to the spectrum of a composition operator $C_φ$, defined by a rotation $φ(z)=rz$ with $|r|=1$, on the space $H_0(\mathbb{D})$ of all analytic functions on the unit disc which vanish at $0$, is given. Examples show that the point $1$ may or may not belong to the spectrum of $C_φ$, and this is related to Diophantine approximation. Our results complement recent work by Arendt, Celariès and Chalendar.
By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner-outer factorization. Here, a bounded analytic function is called \emph{inner} or \emph{outer} if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, $H^2$, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function $θ$ decomposes uniquely as the product of a \emph{Blaschke inner} function and a \emph{singular inner} …
Analytic automorphism group and similar representation of analytic functions
本文受 Gromov 多项式增长定理启发，研究多项式增长的加权 Hardy 空间，证明解析自同构诱导的复合算子有界且 M_z 与 M_φ 相似，并肯定回答 Douglas 问题的广义版本，同时给出中间增长情形下的无界反例及 Jordan 表示与相似分类。
Operator-valued multivariable Bohr type inequalities are obtained for: a class of noncommutative holomorphic functions, generalizing the analytic functions on the open unit disc; the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra; a class of noncommutative selfadjoint harmonic functions, generalizing the real-valued harmonic functions on the open unit disc; Cuntz-Toeplitz algebras ; the reduced (resp.full) group C*-algebra of the free group with n generators; a class of analytic functions on the open unit ball of C^n. The classical Bohr inequality is shown to be a…
Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions in the open unit disc. We study the continuity of $\A$ with respect to various norms. Our main result is that, for a class of norms satifying certain natural axioms, $\A$ is continuous at any function whose superoptimal singular values are non-zero and is such that certain associated integer indices are equal to 1. We also obtain necessary conditions for cont…
The paper proposes representation functionals in a dual paradigm where learning jointly concerns both linear convolutional weights and parametric forms of nonlinear activation functions. The nonlinear forms proposed for performing the functional representation are associated with a new class of parametric neural transfer functions called rectified power sigmoid units. This class is constructed to integrate both advantages of sigmoid and rectified linear unit functions, in addition with rejecting the drawbacks of these functions. Moreover, the analytic form of this new neural class involves sca…
In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna-Pick interpolation problem for analytic functions with positive real parts on the open unit disc. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball on C^n.
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