On quasinormal,subnormal, and hyponormal Toeplitz operators

Let ? be a non-constant function inL ^∞(D) such thatφ=φ ₁+φ ₂, whereφ ₁ is an element in the Bergman spaceL _a ² (D), and \(\phi _2 \in \overline {L_a^2 (D)} \) , the space of all complex conjugates of functions inL _a ² (D). In this paper, it is shown that if 1 is an element in the closure of the range of the self-commutator ofT _?, \(T_{\bar \phi } T_\phi - T_\phi T\phi \) , then the Toeplitz operatorT _? defined ofL _a ² (D) is not quasinormal. Moreover, if \(\phi = \psi + \lambda \bar \psi \) , whereψ∈ H ^∞(D), and λεC, it is proved that ifT _? is quasinormal, thenT _? is normal. Also, the spectrum of a class of pure hyponormal Toeplitz operators is determined.     

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ? ⁿ . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L ² estimates for the $\bar{\partial}$ operator are key ingredients in the proof.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω?C, and in Fock spacesA ²(C ^N),N≧1.     

The Essential Norm of Operators on {A^p_\alpha(\mathbb{B}_n)}

In this paper we characterize the compact operators on the weighted Bergman spaces ${A^p_\alpha(\mathbb{B}_n)}$ when 1 < p < ∞ and α > ?1. The main result shows that an operator on ${A^p_\alpha(\mathbb{B}_n)}$ is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.     

Some integral operators acting on H ∞

Let f and g be analytic on the unit disk \({\mathbb{D}}\) . The integral operator T _g is defined by \({ T_g f(z) = \int_0^z f(t)g'(t) \,dt, z \in \mathbb{D}}\) . The problem considered is characterizing those symbols g for which T _g acting on H ^∞, the space of bounded analytic functions on \({\mathbb{D}}\) , is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, \({ S_g f(z)= \int_0^z f'(t)g(t) \,dt}\) , acting on H ^∞ are also studied.     

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

In 1999 Nina Zorboska and in 2003 P. S.Bourdon, D. Levi, S.K.Narayan and J.H. Shapiro investigated the essentially normal composition operator ${C_\varphi }$ , when φ is a linear-fractional self-map of D. In this paper first, we investigate the essential normality problem for the operator T _w ${C_\varphi }$ on the Hardy space H ², where w is a bounded measurable function on ?D which is continuous at each point of F(φ), φ ∈ S(2), and T _w is the Toeplitz operator with symbol w. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on H ².     

Commutative properties of singularly perturbed operators

Suppose the self-adjoint operatorA in the Hilbert spaceH commutes with the bounded operatorS. Suppose another self-adjoint operatorā is singularly perturbed with respect toA, i.e., it is identical toA on a certain dense set inH. We study the following question: Under what conditions doesā also commute withS? In addition, we consider the case whenS is unbounded and also the case whenS is replaced by a singularly perturbed operator S. As application, we consider the Laplacian inL ₂(R ^q ) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inR ^q ,q=2, 3, or with regular representations of arbitrary isometric transformations inR ^q ,q≤3.     

Generalized sum of operators

For a semibounded below self-adjoint operatorA in a Hilbert spaceH and a singular operatorV acting in theA-scale of Hilbert spaces, the notion of generalized sumA?V is introduced. Conditions are found forA?V to be self-adjoint in ?. In particular, it is shown that if a symmetric operatorV is semibounded or has a spectral gap, then there exists an α such that the generalized sumA?αV is a self-adjoint operator inH. For a symmetric restrictionA = A‖D, D C D(A), with deficiency indices (1, 1), it is proved that each self-adjoint extension à of A admits representation as a generalized sum Ã=A?V.     

Banach-valued holomorphic functions on the maximal ideal space of H ∞

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H ^∞ of bounded holomorphic functions on the unit disk $\mathbb{D}\subset \mathbb{C}$ with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H ^∞, prove that the maximal ideal space of the algebra $H_{\mathrm{comp}}^{\infty}(A)$ of holomorphic functions on $\mathbb{D}$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H ^∞ and A.     

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.     

Unitary Equivalence, Similarity and Calculation of K0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.     

Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem

A functionG in a Bergman spaceA ^p , 0<p<∞, in the unit diskD is calledA ^p -inner if |G| ^p ?1 annihilates all bounded harmonic functions inD. Extending a recent result by Hedenmalm forp=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ _D denotes the characteristic function ofD andG is an arbitraryA ^p -inner function, is continuous inC, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman spaceA ² to a generalA ^p -setting.     

Norm convergence of normalized iterates and the growth of Kœnigs maps

Let ? be an analytic function defined on the unit diskD, with ?(D)?D, ?(0)=0, and ?′(0)=λ≠0. Then by a classical result of G. K?nigs, the sequence of normalized iterates Φ _n /λ ⁿ converges uniformly on compact subsets ofD to a function σ analytic inD which satisfiesσ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy spaceH _p , the sequence Φ _n /λ ⁿ converges to σ in the norm ofH _p . We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ? is univalent. When ? is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to anyH _p . It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman spaceL _a ^p .     

Spectral theory for operator matrices related to models in mechanics

We derive various properties of the operator matrix where A ₀ is a uniformly positive operator and A ₀ ^?1/2 DA ₀ ^?1/2 is a bounded nonnegative operator in a Hilbert space H. Such operator matrices are associated with second-order problems of the form $ \ddot z(t) + A_0 z(t) + D\dot z(t) = 0 $ , which are used as models for transverse motions of thin beams in the presence of damping.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

On an estimate of Calderón-Zygmund operators by dyadic positive operators

Given a general dyadic grid D and a sparse family of cubes S = {Q _j ^k ∈ D, define a dyadic positive operator A _D,S by $${A_{D,S}}f(x) = \sum\limits_{j,k} {{f_{Q_j^k}}{\chi _{Q_j^k}}} (x)$$ . Given a Banach function space X(? ⁿ ) and the maximal Calderón-Zygmund operator ${T_\natural }$ , we show that $${\left\| {{T_\natural}f} \right\|_X} \leqslant c(T,n)\mathop {\sup }\limits_{D,S} {\left\| {{A_{D,S}}|f|} \right\|_X}$$ This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A ₂ conjecture”; (iii) an extension of certain mixed A _p ?A _r estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A ₁ estimates (known for T ) to the maximal Calderón-Zygmund operator $\natural $ .