Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

On inner functions with [formula omitted] and [formula omitted] derivatives in the unit ball of [formula omitted

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Toeplitz operators on weighted Bergman spaces on finitely connected domains

We consider Toeplitz operators $T_f^{\lambda}$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\Omega\subset \mathbb{C}^n$. Here $\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{\lambda \rightarrow \infty} \big{\|}T_f^{\lambda} T_g^{\lambda} -T_{fg}^{\lambda} \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $\Omega=\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\Omega$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\beta$ denote the Bergman metric distance function on $\Omega$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(\Omega)$ of all $\beta$-uniformly continuous functions on $\Omega$. Note that this space contains also unbounded functions. In case of the complex unit ball $\Omega=\mathbb{B}^n \subset \mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\rm VMO}(\mathbb{B}^n)$, where the vanishing oscillation inside $\mathbb{B}^n$ is measured with respect to $\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\Omega$.

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.

The aim of this paper is to show that univalent functions in several classical function spaces can be characterized by integral conditions involving the maximum modulus function. For a suitable choice of parameters, the established condition or its appropriate variant reduces to a known characterization of univalent functions in the Hardy or weighted Bergman space and gives a new characterization of univalent functions in several Mobius invariant function spaces, such as BMOA, Qp or the Bloch space. It is proved, for example, that univalent functions in the Dirichlet type space \( \mathcal{D}_{p + \alpha }^p \) are the same as the univalent functions in Hαp and Sαp if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller Mobius invariant subspace of the Bloch space than Qp still containing all univalent Bloch functions.

The commutative C*-algebras of Toeplitz operators on the classical (weightless) Bergman space were classified in Chapter 9 by pencils of geodesics on the unit disk, considered as the hyperbolic plane. Theorem 10.4.1 shows that the same classes of defining symbols generate commutative C*-algebras of Toeplitz operators on each weighted Bergman space. At the same time the principal question, whether the above cases are the only possible sets of defining symbols which might generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space, has remained open.

Is the Ordering of Imaging for Suspected Venous Thromboembolism Consistent With D-dimer Result?

A set E c Q is holomorphically dominating for Q if sUP,cE If(z)I = SUpzcn If(z)l for all holomorphic functions f on Q. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point * (of Q) from Q x E. Moreover, it is equivalent to a large number of other statements (old and new) of holomorphic, harmonic and topological nature, including that a certain weighted Bergman space with p = oo is a Banach space. We extend this to the cases of harmonic functions in R' and holomorphic functions in cV. We also present some results on when weighted Bergman spaces are (quasi)-Banach spaces, the case p = oo being characterised by the result mentioned above.

We study the structure of inductive limits of weighted spaces of harmonic and holomorphic functions defined on the open unit disk of ℂ, and of the associated weighted locally convex spaces. Using a result of Lusky we prove, for certain radial weights on the open unit disk D of ℂ, that the spaces of harmonic and holomorphic functions are isomorphic to complemented subspaces of the corresponding Köthe sequence spaces. We also study the spaces of harmonic functions for certain non-radial weights on D. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on ∂D.

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.

The 13C-urea breath test for diagnosis of Helicobacter pylori infection has not been validated in infants and young children. The influence of age on the test results was studied by conventional validation against invasive methods and by mathematical estimation in a large pediatric population. The breath test was performed in 1499 children aged 2 months to 18 years. After a fasting period of 4 hours or more, 75 mg 13C-urea was ingested with cold apple juice, breath samples were taken at baseline and at 15 and 30 minutes. The distribution of the natural logarithms of the delta-over baseline (DOB) values were calculated, and the optimal cutoff values between positive and negative test results and gray zones with a risk of misclassification more than 10% were determined for both time points. In a subgroup of 149 children results of the breath test were compared with concordant results of histology and rapid urease test; 53 of them were less than 6 years of age. Logarithmic results of 1499 breath tests revealed two normally distributed subgroups with minimal overlap. The calculated optimal cutoff values were 4.7/1000 at 15 minutes and 5.0/1000 at 30 minutes. At 30 minutes, only 2.6% of all results were in the calculated gray zone (2.6-6.5/1000). Age was negatively correlated to DOB values of both negative (r = -0.223) and positive results (r = -0.291; P < 0.001). Breath test-negative and -positive children 6 or less years of age had significantly higher mean DOB values (P < 0.02) and a larger proportion of results within the gray zone than older children. Compared with biopsy-based results, the least discrepancies occurred at a cutoff of 5.0/1000: 0 of 61 infected (sensitivity 100%) and 6 of 88 noninfected children. Because five of the false-positive results were obtained in children less than 6 years of age, specificity and positive predictive values were lower in this age group than in older patients (88.1% vs. 97.8% and 68.8% vs. 98.0%, respectively). Under the applied conditions, the 13C-urea breath test shows an excellent separation between positive and negative results. Because of some overlap and a strong age effect, definition of a gray zone appears more meaningful than a threshold value. Because infants and young children have a high risk for false-positive breath test results, the values for cutoff and gray zones may have to be adapted. Further validation studies against invasive methods are warranted in this age group.

Exploring the scientific impact of negative results

Let H2(D) denote the Hardy space of a bounded symmetric domain $$D \subset \mathbb{C}^n $$ in its standard Harish-Chandra realization, and let $$A_\alpha ^p (D)$$ be the weighted Bergman space with $$p \geq 1$$ and $$\alpha < \varepsilon _D ,$$ where $$\varepsilon _D $$ is a critical value depending on D. Suppose that $$\phi :D \to D$$ is holomorphic. We show that if the composition operator $$C_\phi $$ defined by $$C_\phi (f) = f \circ \phi $$ is compact (or, more generally, power-compact) on H2(D) or $$A_\alpha ^p (D),$$ then ϕ has a unique fixed point z0 in D. We then prove that the spectrum of $$C_\phi $$ as an operator on these function spaces is precisely the set consisting of 0, 1, and all possible products of eigenvalues of $$\phi '(z_0 ).$$ These results extend previous work by Caughran/Schwartz and MacCluer. As a corollary, we now have that MacCluer’s previous spectrum results on the unit ball B n extend to H p (Δn) (not only for p = 2 but for all p > 1) and $$A_\alpha ^p (\Delta ^n )$$ (for p ≥ 1), where Δ n is the polydisk in $$\mathbb{C}^n .$$

We discuss new results concerning unbounded Toeplitz operators defined in Segal-Bargmann spaces of (vector-valued) functions, i.e. the space of all entire functions which are square summable with respect to the Gaussian measure in $\mathrm{C}^n$. The problem of finding adjoints of analytic Toeplitz operators is solved in some cases. Closedness of the range of analytic Toeplitz operators is studied. We indicate an example of an entire function inducing a Toeplitz operator, for which the space of polynomials is not a core though it is contained in its domain.