The operator-valued Poisson kernel and its applications (original) (raw)

On analytic families of operators

Israel Journal of Mathematics, 1969

The classical Riesz-Thorin interpolation theorem [6] was extended by Hirschman [2] and Stein [5] to analytic families of operators. We recall the notions: Let F(z), z = x+iy, be analytic in 0< Re z< 1 and continuous in 0 =< Re z _< 1. F(z) is said to be of admissible growth iff Sup log iF(x + iY) I < Ae~Iyl where a < 7z. O<_x~l The significance of this notion is in the following lemma due to Hirschman [2]: LEMMA. lf F(z) is of admissible growth and ifloglF(it) l ~ ao(t), log IF(i+ it) I <= a~(t) then log I F(0) ]_<f_% Po(0, t)ao(t)dt + f 2~ P~(O, t)a~(t)dt where P~(O, t) are the values of the Poisson kernel Jor the strip, on Rez = 0, Rez = 1. We next define analytic families of linear operators: Let (M,/~) (N, v) be two measure spaces. Let {~} be a family of linear operators indexed by z, 0 ~ Re z ~ 1 so that for each z, Tz is a mapping of simple functions on M to measurable functions on N. {T~} is called an analytic family iff for any measurable set E of M of finite measure, for almost every y 6 N, the function qSr(z) = T~(X~)(y) is analytic in 0 < Re z < 1, continuous in 0 ~ Re z __< 1. The analytic family is of admissible growth iff for almost every y ~ N, ~by(z) is of admissible growth. We finally recall the notion of L(p, q) spaces. An exposition of these spaces can be found in Hunt [3]. Let f be a complex valued measurable function defined on a ~-finite measure space (M,/~). # is assumed to be non-negative. We assume that f is finite valued a.e., and denoting Ey = {x/If(x)] > Y}, 2r(y) = /~(Ey), we assume also that for some y > 0, 2;(y)< oo. We define f*(t) = Inf{y > Oily(y) < t}.

Operators on some analytic function spaces and their dyadic counterparts

In this thesis we consider several questions on harmonic and analytic functions spaces and some of their operators. These questions deal with Carleson-type measures in the unit ball, bi-parameter paraproducts and multipliers problem on the bitorus, boundedness of the Bergman projection and analytic Besov spaces in tube domains over symmetric cones. In part I of this thesis, we show how to generate Carleson measures from a class of weighted Carleson measures in the unit ball. The results are used to obtain boundedness criteria of the multiplication operators and Ces`aro integral-type operators between weighted spaces of functions of bounded mean oscillation in the unit ball. In part II of this thesis, we introduce a notion of functions of logarithmic oscillation on the bitorus. We prove using Cotlar’s lemma that the dyadic version of the set of such functions is the exact range of symbols of bounded bi-parameter paraproducts on the space of functions of dyadic bounded mean oscillatio...

Semigroup Kernels, Poisson Bounds, and Holomorphic Functional Calculus

Journal of Functional Analysis, 1996

Let L be the generator of a continuous holomorphic semigroup S whose action is determined by an integral kernel K on a scale of spaces L p (X ; \). Under mild geometric assumptions on (X, \), we prove that if L has a bounded H-functional calculus on L 2 (X; \) and K satisfies bounds typical for the Poisson kernel, then L has a bounded H-functional calculus on L p (X ; \) for each p # (1,). Moreover, if (X, \) is of polynomial type and K satisfies second-order Gaussian bounds, we establish criteria for L to have a bounded Ho rmander functional calculus or a bounded Davies Helffer Sjo strand functional calculus.

Mini-Workshop: Operators on Spaces of Analytic Functions

Oberwolfach Reports, 2000

The major topics discussed in this workshop were invariant subspaces of linear operators on Banach spaces of analytic functions, the ideal structure of H ∞ , asymptotics for condition numbers of large matrices, and questions related to composition operators, frequently hypercyclic operators, subnormal operators and generalized Cesàro operators. A list of open problems raised at this workshop is also included.

Hankel operators and the Stieltjes moment problem

Journal of Functional Analysis, 2010

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it. We denote by μ n the image measure in C n of μ ⊗ σ n under the map (t, ξ) → √ tξ, where σ n is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L 2 (μ n) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n = 1, we prove that there are nontrivial Hilbert-Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting.

Operator-valued Nevanlinna-Pick kernels and the functional models for contraction operators

Integral Equations and Operator Theory, 1987

Given a contractive operator function B(z) analytic on the unit disk, de Branges and Rovnyak introduced the Hilbert space ~(B) of vector-valued analytic functions with reproducing kernel KB(w,z) ~ (l-wz)-l(I-B(z)B(w)*). We obtain a necessary and sufficient condition for one such space ~(B2) to be contained in another ~(BI), or equivalently, for the existence of a positive number a for which the B I operator-valued kernel aK-K B2 is positive semidefinite. The condition involves the outer factor A of the maximal factorable minorant A A of I-B B. We show that the space ~(B) is naturally isomorphic to the Sz.-Nagy-Foias model space IK(8) where 8 is the contractive matrix function and then solve the above inclusion problem as an application of the lifting theorem of Sarason, Sz.-Nagy and Foias. Along the way we delineate the role of A in the fine structure of the spaces ~(B) and IK(B), and give a partial characterization of the extreme points in the unit ball of operator-valued H ~. INTRODUCTION. Let ~ and ~ be complex separable Hilbert spaces and let B be an analytic function Jn the unit IThis research supported in part by the National Science Foundation.

Spectral Theorems for a Class of Toeplitz Operators on the Bergman Space

Houston Journal of Mathematics, 1986

Let G be a countable union of annuli centered at 0 and contained in the unit disc D; let 1G be the characteristic function of G. Let T G be the Toeplitz operator on the Bergman space A 2 with symbol 1G. We show that the essential spectrum of T G is connected, and we give upper and lower bounds on the spectrum and essential spectrum in terms of the radii of the annuli. 1. Introduction. Let C be the complex plane and let D be the open unit disc. Let dA be normalized (Lebesgue) area measure on D; define Løø(D) as the complex-valued functions defined on D which are measurable and essentially bounded with respect to dA. Define L2(D) to be the complex-valued functions defined on D which are measurable and square-integrable with respect to dA. L2(D) is a Hilbert _ space with inner product (f,g)L2 = fD f fg dA. Define the Bergman space A 2 to be the analytic functions in L2(D). It is well known that A 2 is a closed subspace of L2(D) and is thus a Hilbert space (see [4]). Let P be the projection from L2(D) onto A 2. Let fC Løø(D); the Toeplitz operator on A 2 with symbol f, denoted Tf, is defined by Tf(g) = P(fg), g C A 2. We shall denote the spectrum of Tf by o(Tf). Let B(A 2) be the Banach algebra of bounded linear operators on A2; let •c(A 2) denote the ideal of compact operators on A 2. The quotient algebra B(A2)/•c(A 2) is referred to as the Calkin algebra, and the quotient map rr: B(A 2)-• B(A2)fic(A 2) is referred to as the Calkin map. The essential spectrum of Tf, denoted oe(Tf), is the spectrum of rr(Tf) in the Calkin algebra. In this work we shall consider Toeplitz operators on A 2 for which the symbol is the characteristic (indicator) function of a measurable subset G of D. An important 397 398 JAMES W. LARK, III application of results concerning these operators is in the work of Voas [8]. Let 1G denote the characteristic function of G; i.e., 1G(Z) = 1 if z E G, 1G(Z) = 0 if z • G. We shall use the symbol T G to represent the operator T1G on A 2. It is easy to see that T G is self-adjoint, and that o(T G) _C [0,1]. Since T G is self-adjoint (and thus rr(TG)is self-adjoint), then sup o(T G) = sup{X: 3, E O(TG)} = I[TGII and sup oe(T G) = sup{X: 3, G oe(TG)} = I[TGlle (the essential norm of TG).

Carleson Measures and Toeplitz Operators

Lecture Notes in Mathematics, 2017

In this last chapter we shall describe a completely different application of the Kobayashi distance to complex analysis. To describe the problem we need a few definitions. Definition 6.0.1. We shall denote by n the Lebesgue measure in C n. If D ⇢⇢ C n is a bounded domain and 1  p  •, we shall denote by L p (D) the usual space of measurable p-integrable complex-valued functions on D, with the norm k f k p =  Z D | f (z)| p dn(z) 1/p if 1  p < •, while k f k • will be the essential supremum of | f | in D. Given b 2 R, we shall also consider the weighted L p-spaces L p (D, b), which are the L p spaces with respect to the measure d b n, where d : D ! R + is the Euclidean distance from the boundary: d (z) = d(z, ∂ D). The norm in L p (D, b) is given by k f k p,b =  Z D | f (z)| p d (z) b dn(z) 1/p for 1  p < •, and by k f k b ,• = k f d b k • for p = •. Definition 6.0.2. Let D ⇢⇢ C n be a bounded domain in C n , and 1  p  •. The Bergman space A p (D) is the Banach space A p (D) = L p (D) \ Hol(D, C) endowed with the norm k • k p. More generally, given b 2 R the weighted Bergman space A p (D, b) is the Banach space A p (D, b) = L p (D, b) \ Hol(D, C) endowed with the norm k • k p,b. The Bergman space A 2 (D) is a Hilbert space; this allows us to introduce one of the most studied objects in complex analysis.