In this paper, we prove the corona theorem for $M(D(mu_k))$ in two different ways, where $mu_k = sum_{i=1}^k a_i delta_{zeta_i}$. Then we prove that the Bass stable rank of $M(D(mu_k))$ is one.
In this paper, we use a new method to solve a long-standing problem. More specifically, we show that the Beurling-type theorem holds in the Bergman space $A^2_alpha(D)$ for any $-1<alpha < +infty$. That is, every invariant subspace $H$ for the shift
operator $S$ on $A^2_alpha(D)$ $(-1<alpha < +infty)$ has the property $H=[Hominus zH]_{S,A^2_alphaleft(Dright)}$.
The well known Douglas Lemma says that for operators $A,B$ on Hilbert space that $AA^*-BB^*succeq 0$ implies $B=AC$ for some contraction operator $C$. The result carries over directly to classical operator-valued Toeplitz operators by simply replacin
g operator by Toeplitz operator. Free functions generalize the notion of free polynomials and formal power series and trace back to the work of J. Taylor in the 1970s. They are of current interest, in part because of their connections with free probability and engineering systems theory. For free functions $a$ and $b$ on a free domain $cK$ defined free polynomial inequalities, a sufficient condition on the difference $aa^*-bb^*$ to imply the existence a free function $c$ taking contractive values on $cK$ such that $a=bc$ is established. The connection to recent work of Agler and McCarthy and their free Toeplitz Corona Theorem is exposited.
In 1975, P.R. Chernoff used iterates of the Laplacian on $mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $mathbb{R}^n$ to be quasi-analytic. In this paper, we prove
an exact analogue of Chernoffs theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.
This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings an
d clarifies the behaviour of these invariants when 2 is not invertible. In this article we lay the foundations of our approach by considering Luries notion of a Poincare $infty$-category, which permits an abstract counterpart of unimodular forms called Poincare objects. We analyse the special cases of hyperbolic and metabolic Poincare objects, and establish a version of Ranickis algebraic Thom construction. For derived $infty$-categories of rings, we classify all Poincare structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincare structures on $infty$-categories of parametrised spectra, recovering the visible signature of a Poincare duality space. We conduct a thorough investigation of the global structural properties of Poincare $infty$-categories, showing in particular that they form a bicomplete, closed symmetric monoidal $infty$-category. We also study the process of tensoring and cotensoring a Poincare $infty$-category over a finite simplicial complex, a construction featuring prominently in the definition of the L- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0-th Grothendieck-Witt group of a Poincare $infty$-category using generators and relations. We extract its basic properties, relating it in particular to the 0-th L- and algebraic K-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications.
Let $(A, m, k)$ be a Gorenstein local ring of dimension $ dgeq 1.$ Let $I$ be an ideal of $A$ with $htt(I) geq d-1.$ We prove that the numerical function [ n mapsto ell(ext_A^i(k, A/I^{n+1}))] is given by a polynomial of degree $d-1 $ in the case w
hen $ i geq d+1 $ and $curv(I^n) > 1$ for all $n geq 1.$ We prove a similar result for the numerical function [ n mapsto ell(Tor_i^A(k, A/I^{n+1}))] under the assumption that $A$ is a CM ~ local ring. oindent We note that there are many examples of ideals satisfying the condition $curv(I^n) > 1,$ for all $ n geq 1.$ We also consider more general functions $n mapsto ell(Tor_i^A(M, A/I_n)$ for a filtration ${I_n }$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal CM ~ $A$-module and ${I_n=overline{I^n} }$ is the integral closure filtration, $I$ an $m$-primary ideal in $A.$