Weighted modular inequalities for Hardy-Steklov operators

We characterize weighted modular inequalities of weak and strong type for the Hardy-Steklov operators T defined by , where g is a positive function and s, h are increasing and continuous functions such that s(x)?h(x) for all x.     

BiBloch mappings and composition operators from Bloch type spaces to BMOA

The problem of constructing functions f₁, f₂ analytic in the unit disc D of the complex plane satisfying

     

Composition operators induced by symbols defined on a polydisk

Suppose φ is a holomorphic mapping from the polydisk Dm into the polydisk Dn, or from the polydisk Dm into the unit ball Bn, we consider the action of the associated composition operator Cφ on Hardy and weighted Bergman spaces of Dn or Bn. We first find the optimal range spaces and then characterize compactness. As a special case, we show that if

     

Toeplitz operators associated to commuting row contractions

Let H be a complex Hilbert space and let {Tn}_n?1 be a sequence of commuting bounded operators on H such that . Let denote the space of all operators X in B(H) for which and suppose that . We will show that there exists a triple {K,Γ,{Un}_n?1} where K is a Hilbert space, Γ:K→H is a bounded operator and {Un}_n?1⊂B(K) is a sequence of commuting normal operators with such that TnΓ=ΓUn for n?1, and for which the mapping Y?ΓYΓ^∗ is a complete isometry from the commutant of {Un}_n?1 onto the space . Moreover we show that the inverse of this mapping can be extended to a ^∗-homomorphism

     

Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces

We prove that if ω, ω₁, ω₂, v₁, v₂ are appropriate, , j=1,2, and ωa∈Lp, then the Toeplitz operator Tph₁,h₂(a) from to belongs to the Schatten-von Neumann class of order p. From this property we prove convolution properties between weighted Lebesgue spaces and Schatten-von Neumann classes of symbols in pseudo-differential calculus.     

Coincidence of strict s-numbers of weighted Hardy operators

We establish the equality of all the so-called strict s-numbers of the weighted Hardy operator T:Lp(I)→Lp(I), where 1<p<∞, I=(a,b)⊂R and

     

Schauder estimates for parabolic nondivergence operators of Hörmander type

Let X₁,X₂,…,Xq be a system of real smooth vector fields satisfying Hörmander's rank condition in a bounded domain Ω of Rn. Let be a symmetric, uniformly positive definite matrix of real functions defined in a domain U⊂R×Ω. For operators of kind

     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Heat flow, BMO, and the compactness of Toeplitz operators

In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1?p<∞ whenever (vanishes at infinity) or , respectively, for some s with , where is the heat transform of g on Cn. Moreover, we show that compactness of Tg implies that is in C₀(Cn) for all and use this to show that, for g∈BMO¹(Cn), we have is in C₀(Cn) for some s>0 only if is in C₀(Cn) for alls>0. This “backwards heat flow” result seems to be unknown for g∈BMO¹ and even g∈L^∞. Finally, we show that our compactness and vanishing “backwards heat flow” results hold in the context of the weighted Bergman space , where the “heat flow” is replaced by the Berezin transform Bα(g) on for α>−1.     

Hyponormality and subnormality for powers of commuting pairs of subnormal operators

Let H₀ (respectively H_∞) denote the class of commuting pairs of subnormal operators on Hilbert space (respectively subnormal pairs), and for an integer k?1 let Hk denote the class of k-hyponormal pairs in H₀. We study the hyponormality and subnormality of powers of pairs in Hk. We first show that if (T₁,T₂)∈H₁, the pair may fail to be in H₁. Conversely, we find a pair (T₁,T₂)∈H₀ such that but (T₁,T₂)∉H₁. Next, we show that there exists a pair (T₁,T₂)∈H₁ such that is subnormal (for all m,n?1), but (T₁,T₂) is not in H_∞; this further stretches the gap between the classes H₁ and H_∞. Finally, we prove that there exists a large class of 2-variable weighted shifts (T₁,T₂) (namely those pairs in H₀ whose cores are of tensor form (cf. Definition 3.4)), for which the subnormality of and does imply the subnormality of (T₁,T₂).     

Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights

We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap(w)⊂Lq(μ), 0<p,q<∞, where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights , α>0, and some double exponential type weights.As an application of that result, we characterize the boundedness and the compactness of Tg:Ap(w)→Aq(w), 0<p,q<∞, w∈W, where Tg is the integration operator

     

Dilation equations and Markov operators

Using the theory of Markov operators, we give a new proof of the known fact saying that for every positive integers N and k>1, and for every nonnegative reals c₀,…,cN satisfying the first sum rule the dilation equation

     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L²(R₊,S) by the differential expression

     

Integral-type operators acting between weighted-type spaces on the unit ball

Let B denote the unit ball in Cⁿ and H(B) the space of all holomorphic functions on B. We study the boundedness and compactness of the following integral-type operators

     

Accretive operators and Cassels inequality

Let a,b>0 and let Z∈M_n(R) such that Z lies into the operator ball of diameter [aI,bI]. Then for all positive definite A∈M_n(R),

     

Maps preserving the spectrum of generalized Jordan product of operators

Let A₁,A₂ be standard operator algebras on complex Banach spaces X₁,X₂, respectively. For k?2, let (i₁,…,i_m) be a sequence with terms chosen from {1,…,k}, and define the generalized Jordan product

     

Integral-type operators from Bloch-type spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball . This paper investigates the following integral-type operator with symbol

where is the radial derivative of g. The boundedness and compactness of the operator T_g from Bloch-type spaces to Zygmund-type spaces are studied.