A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

Generalization of certain subclasses of multivalent functions with negative coefficients defined by using a differential operator

Making use of a differential operator, we introduce and study a certain class SC_n(j,p,q,α,λ) of p-valently analytic functions with negative coefficients. In this paper, we obtain numerous sharp results including (for example ) coefficient estimates, distortion theorem, radii of close-to convexity, starlikeness and convexity and modified Hadamard products of functions belonging to the class SC_n(j,p,q,α,λ). Finally, several applications investigate an integral operator, and certain fractional calculus operators are also considered.     

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P_λ^(1/g) (χ₁, …, χ_n) …, χ_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g) (x₁, …, x_k, 1, … 1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.     

Minimal Extensions in Tensor Product Spaces

LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM (S, V),N (Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλ_MN(V_αW,X_αY)=λ_M(V, X) λ_N(W, Y).Here for any Banach spaceX,VSXandM (S, V)

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Also some applications of the above mentioned result will be presented.     

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Pták defined the notion of generalized Hankel operator as follows: Given two contractions and , an operatorX: is said to be a generalized Hankel operator ifT ₂ X=XT ₁ ^* andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT ₁ andT ₂. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Pták's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.Dedicated to the memory of our master and friend Vlastimil Pták     

On a quotient norm and the Sz.-Nagy-Foiaş lifting theorem

It is shown that the infimum over all choices of the operator X of the norm of the operator matrix [ ], whose entries are operators on Hilbert spaces, is the minimum of the norms of the first row and of the first column, and an explicit formula for a minimizing X is given in terms of A, B, C, and their adjoints. A generalization of a fundamental theorem on Hankel operators is seen to follow immediately from this result. The formula is then used to prove a generalization of the Sz. Nagy-Foiaş lifting theorem which in turn yields interpolation theorems for analytic functions from the unit disc to a von Neumann algebra. The generalized lifting theorem also implies a generalization of the theorem of Ando asserting the existence of commuting unitary dilations for a pair of commuting contractions and a generalized von Neumann inequality ∑ a_jkS^jT^k sup{ ∑ A_jkz^jw^k ¦ ¦ z ¦ = ¦ w ¦ = 1} for operator polynomials ∑ A_jkS^jT^k in two commuting contractions S, T with operator coefficients A_jk which commute with S, T and their adjoints.     

On the Spectrum of Frobenius–Perron Operators

We prove more results on the spectrum of the Frobenius–Perron operator P: L¹ → L¹ associated with a nonsingular transformation S: X → X on a σ-finite measure space (X, Σ, μ).     

Operators of Hankel type

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered. The present note provides a parametric labeling of all the Hankel symbols of a given Hankel operator X by means of Schur class functions. The result includes uniqueness criteria and a Schur like formula. As a by-product, a new proof of the existence of Hankel symbols is obtained. The proof is established by associating to the data of the problem a suitable isometry V so that there is a bijective correspondence between the symbols of X and the minimal unitary extensions of V.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Abstract Hankel Operators in Kreĭn Spaces

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered in the Kreĭn space setting. Under a generic assumption, without which the Krein space case may be untreatable, a necessary and sufficient condition for the existence of Hankel symbols for a given Hankel operator X is given. A parametric labeling of the Hankel symbols of X by means of Schur class functions is obtained. The proof is established by associating to the data of the problem an isometry V acting on a Kreĭn space so that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of V . The result includes uniqueness criteria and a Schur like formula.     

The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (x_n) → x in X and (x_n*) → 0 in X* with ||x_n|| = ||x||= 1 we have (x_n*(x_n)) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S ↦ Sh, S ↦ S^th are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.     

Energy dependence of the scattering operator

We prove that the scattering operator S(E) depends continuously on the energy E for a certain class of Schrodinger operators, by an abstract method using trace conditions and the dilation group. We also obtain pointwise bounds on S(E) −1₁as E → ∞, and even as E → 0 in the case of repulsive potentials.     

A bijective proof of the hook-length formula

A well-known theorem of Frame, Robinson, and, Thrall states that if λ is a partition of n, then the number of Standard Young Tableaux of shape λ is n! divided by the product of the hook-lengths. We give a new combinatorial proof of this formula by exhibiting a bijection between the set of unsorted Young Tableaux of shape λ, and the set of pairs (T, S), where T is a Standard Young Tableau of shape λ and S is a “Pointer” Tableau of shape λ.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

Given two self-adjoint operators A and V = V₊  − V₋ , we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N₊ (V; λ, α), N₋ (V; λ, α), and N₀(V; λ, α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α → ∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators.     

Embeddings of Lorentz spaces of vector-valued martingales

We obtain new embedding theorems for Lorentz spaces of vector-valued martingales, thus generalizing the classical martingale inequalities. In contrast to earlier methods, we use martingale transformations defined by sequences of operators and identify the operator S ^(p)(f) for a martingale f ranging in a Banach space X with the maximal operator for some ℓ ^p (X)-valued martingale transform. The obtained inequalities are closely related to geometric properties of the Banach space in question.     

Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.     

Spectral properties of some Jacobi matrices with double weights

We study the spectral properties of Jacobi matrices with the weights satisfying λ_2n−1=λ_2n=n^a or λ_2n=λ_2n+1=n^a, a>0. We show that for a=1 these are cases of spectral phase transitions in a. We use a new method of estimating transfer matrix products to describe the absolutely continuous part of these operators. For a=1 the existence of a spectral gap is proved. We also show how the results for double weights can be used for the spectral analysis of the Jacobi matrices related to some birth and death processes, previously studied by Janas and Naboko.     

Multiplicativity factors for seminorms. II

Let S be a seminorm on an algebra . In this paper we study multiplicativity and quadrativity factors for S, i.e., constants μ > 0 and λ > 0 for which S(xy) μS(x)S(y) and S(x²) λS(x)² for all x, y A. We begin by investigating quadrativity factors in terms of the kernel of S. We then turn to the question, under what conditions does S have multiplicativity factors if it has quadrativity factors? We show that if is commutative then quadrativity factors imply multiplicativity factors. We further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05