Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

On the equivalence of differential operators of infinite order in analytic spaces

It is shown that in the spacesA _R (0 <R ⩽ ∞) of all functions which are single-valued and analytic in the disk |z| < R with the topology of compact convergence, the differential operator of infinite order with constant coefficients is equivalent to the operator Dⁿ (n is a fixed natural number) if and only if and |ϕ _n | = 1 for R < ∞ or ϕ _n ≠ 0 for R = ∞. Also the equivalence of two shift operators in the space A_∞ is investigated. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 33–37, January, 1977.     

STRONGLY CONTINUOUS INTEGRATED COSINE OPERATOR FUNCTIONS WITH GROWTH ω

For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.     

Around Operator Monotone Functions

We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f(A+B) ￡ f(A)+f(B){f(A+B)\leq f(A)+f(B)} for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f °g{f \circ g} of an operator convex function f on [0,∞) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and present some applications.     

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Generation of cosine families via Lord Kelvin’s method of images

We show that generation theorems for cosine families related to one-dimensional Laplacians in C[0, ∞] may be obtained by Lord Kelvin’s method of images, linking them with existence of invariant subspaces of the basic cosine family. This allows us to deal with boundary conditions more general than those considered before (Bátkal and Engel in J Differ Equ 207:1–20, 2004; Chill et al. in Functional analysis and evolution equations. The Günter Lumer volume, Birkhauser, Basel, pp 113–130, 2007; Xiao and Liang in J Funct Anal 254:1467–1486, 2008) and to give explicit formulae for transition kernels of related Brownian motions on [0, ∞). As another application we exhibit an example of a family of equibounded cosine operator functions in C[0, ∞] that converge merely on C ₀(0, ∞] while the corresponding semigroups converge on the whole of C[0, ∞].     

Multiplication properties of the spacesB

Summary Let A be either B _p, ^qs or F _p, ^qs , where - ∞<s <∞; 0<p, q≦∞ (spaces of Besov-Hardy-Sobolev type, defined on R_n). (i) If g ∈C _ϱ (H?lder-Zygmund spaces), then f → gf is a bounded operator from A into A, provided that ϱ=ϱ(s, p, q, n) is large enough. (ii) There are given sufficient conditions for s, p, and q ensuring that A is a subalgebra of C (space of uniformly continuous bounded functions on R_n). Entrata in Redazione il 17 marzo 1976.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Eigenvalue inequalities for convex and log-convex functions

We give a matrix version of the scalar inequality f(a + b) ? f(a) + f(b) for positive concave functions f on [0, ∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic-geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia-Kittaneh arithmetic-geometric mean inequality.     

K-Theory of pseudodifferential operators with semi-periodic symbols

Let 𝔄 denote the C^*-algebra of bounded operators on L ² ℝ generated by: (i) all multiplications a(M) by functions a∈C[ − ∞, + ∞], (ii) all multiplications by 2π-periodic continuous functions, and (iii) all operator of the form F ⁻¹ b(M)F, where F denotes the Fourier transform and b∈C[ − ∞, + ∞]. A given A ∈ 𝔄 is a Fredholm operator if and only if σ(A) and γ(A) are invertible, where σ denotes the continuous extension of the usual principal symbol, while γ denotes an operator-valued “boundary principal symbol” (the “boundary” here consists of two copies of the circle, one at each end of the real line). We give two proofs of the fact that K ₀(𝔄) is isomorphic to ℤ and that K ₁(𝔄) is isomorphic to ℤ ⊕ ℤ . We do it first by computing the connecting mappings in the six-term exact sequence associated to σ. For the second proof, we show that the image of γ is isomorphic to the direct sum of two copies of the crossed product , where α denotes the translation-by-one automorphism. Its K-theory can be computed using the Pimsner–Voiculescu exact sequence, and that information suffices for the analysis of the standard cyclic exact sequence associated to γ. Received: February 2006     

Extension of smooth functions from finitely connected planar domains

Consider the Sobolev space W _∞ ^k (Ω) of functions with bounded kth derivatives defined in a planar domain. We study the problem of extendability of functions from W _∞ ^k (Ω) to the whole ℝ² with preservation of class, i.e., surjectivity of the restriction operator W _∞ ^k (ℝ²) → W _∞ ^k (Ω).     

Description of sequences of zeros of one class of functions analytic in a half-plane

We describe sequences of zeros of functions ƒ ≠ 0 that are analytic in the right half-plane and satisfy the condition |ƒ(z)| ≤ 0(1) exp (σ| z |η(| z |)), 0 ≤ <+ ∞, Re z > 0, where η: [0; + ∞) → (- ∞; + ∞) is a function of bounded variation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1169–1176, September, 1998.     

On the partially microlocal hypoellipticity for a class of totally characteristic operators with characteristics of constant multiplicity

In this paper, I study the microlocal hypoellipticity for a class of totally characteristic operators (1.1). My main result is as follows: Under the conditions (I), (II), if the indicial operator of (1.1) is microlocally hypoelliptic in the complement ofWF _x(Pu(t,·)) for anyu(t,x)∈C _b ^∞ ([0,T], ℰ),t∈[0,T], λ∈ℤ, then the operator (1.1) is microlocally hypoelliptic in the variablex. Supported by the Natural Science Foundation and Young Men's Science Foundation of Academia Sinica     

On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L _∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L _∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L _∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L _∞(μ) to c ₀(Γ) is narrow while not every such an operator is AM-compact.     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

Cesàro-type operators on Hardy, BMOA and bloch spaces

Let H ^p, p ∈ (0, ∞], BMOA and B ^a, a ∈ (0, ∞) be the classical p-Hardy, analytic BMO(∂) (bounded mean oscillation on the unit circle) and a-Bloch space on the unit disk. In this paper, we prove that the Cesàro-type operator: C ^α, α ∈ (−1, ∞) is bounded on H ^p, p ∈ (0, ∞) and on B ^a, a ∈ (1, ∞), but, unbounded on H ^∞, BMOA and B ^a, a ∈ (0, 1]. In particular, we give an answer to the Stempak’s open problem.     

Asymptotic Behaviour of Convolution Semigroups

In this paper, necessary and sufficient conditions are given for U * μ^*n to converge uniformly on the real axis; here $\mu$ is a nonsingular probability measure on ℝ, and U is a Banach space valued L^∞-function. A connection to uniform convergence of Cesaro mean values is shown. By applying the results to extended orbits of bounded C_0-semigroups on a Banach space X one can relate both kernel and range of the respective generator with those of the derivative operator on L^∞(X). Ergodic theorems and consequences for subordinated semigroups, in particular for holomorphic semigroups, are deduced.     

On analytic solutions of operator differential equations

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation y′(t) = Ay(t), t ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.     

H ∞ -calculus for the Stokes operator on L q -spaces

 It is proved that the Stokes operator on a bounded domain, an exterior domain, or a perturbed half-space Ω admits a bounded H ^∞ -calculus on L _q (Ω) if q(1,∞). Received: 25 January 2002; in final form: 2 October 2002 / Published online: 16 May 2003     

Functional Gabor frame multipliers

A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L^∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L^∞(ℝ) such that {wg_mn} (resp. ωψ_k,ℝ) is a normalized tight frame.