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Register a new userOperator theory and function theory in Drury-Arveson space and its quotients
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Orr Shalit
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The Drury-Arveson space $H^2_d$, also known as symmetric Fock space or the $d$-shift space, is a Hilbert function space that has a natural $d$-tuple of operators acting on it, which gives it the structure of a Hilbert module. This survey aims to introduce the Drury-Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.
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The corona problem was motivated by the question of the density of the open unit disk D in the maximal ideal space of the algebra, H1(D), of bounded holomorphic functions on D. In this note we study relationships of the problem with questions in operator theory and complex geometry. We use the framework of Hilbert modules focusing on reproducing kernel Hilbert spaces of holomorphic functions on a domain, in Cm. We interpret several of the approaches to the corona problem from this point of view. A few new observations are made along the way. 2012 MSC: 46515, 32A36, 32A70, 30H80, 30H10, 32A65, 32A35, 32A38 Keywords: corona problem, Hilbert modules, reproducing kernel Hilbert space, commutant lifting theorem 1
This paper is devoted to the study of reducing subspaces for multiplication operator $M_phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $phi^{-1}circphi$. By this means, we determine the reducing subspaces of $M_phi$ on the Dirichlet space when the order of $phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang cite{DPW12}.
We review recent progress in operator algebraic approach to conformal quantum field theory. Our emphasis is on use of representation theory in classification theory. This is based on a series of joint works with R. Longo.
Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.
We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular contractions ${T_t}_{t>0}$ on $L_p(Omega)$ for a fixed $1<p<infty$. We prove that if a Banach space $X$ is of martingale cotype $q$, then there is a constant $C$ such that $$ left|left(int_0^inftybig|tfrac{partial}{partial t}P_t (f)big|_X^q,frac{dt}tright)^{frac1q}right|_{L_p(Omega)}le C, big|fbig|_{L_p(Omega; X)},, quadforall, fin L_p(Omega; X),$$ where ${P_t}_{t>0}$ is the Poisson semigroup subordinated to ${T_t}_{t>0}$. Let $mathsf{L}^P_{c, q, p}(X)$ be the least constant $C$, and let $mathsf{M}_{c, q}(X)$ be the martingale cotype $q$ constant of $X$. We show $$mathsf{L}^{P}_{c,q, p}(X)lesssim maxbig(p^{frac1{q}},, pbig) mathsf{M}_{c,q}(X).$$ Moreover, the order $maxbig(p^{frac1{q}},, pbig)$ is optimal as $pto1$ and $ptoinfty$. If $X$ is of martingale type $q$, the reverse inequality holds. If additionally ${T_t}_{t>0}$ is analytic on $L_p(Omega; X)$, the semigroup ${P_t}_{t>0}$ in these results can be replaced by ${T_t}_{t>0}$ itself. Our new approach is built on holomorphic functional calculus. Compared with all the previous, the new one is more powerful in several aspects: a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; b) it yields the optimal orders of growth on $p$ for most of the relevant constants; c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood-Paley-Stein inequalities for symmetric submarkovian semigroups are better than the previous by Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when $X$ is of martingale cotype $q$ and ${P_t}_{t>0}$ is the classical Poisson and heat semigroups on $mathbb{R}^d$.
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