A characterization of k-hyponormality via weak subnormality

An operator T acting on a Hilbert space is said to be weakly subnormal if there exists an extension acting on such that for all . When such partially normal extensions exist, we denote by m.p.n.e.(T) the minimal one. On the other hand, for k?1, T is said to be k-hyponormal if the operator matrix is positive. We prove that a 2-hyponormal operator T always satisfies the inequality T∗[T∗,T]T?‖T‖2[T∗,T], and as a result T is automatically weakly subnormal. Thus, a hyponormal operator T is 2-hyponormal if and only if there exists B such that BA∗=A∗T and is hyponormal, where A:=[T∗,T]1/2. More generally, we prove that T is (k+1)-hyponormal if and and only if T is weakly subnormal and m.p.n.e.(T) is k-hyponormal. As an application, we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix.     

On a class of J-self-adjoint operators with empty resolvent set

In the present paper we investigate the set ΣJ of all J-self-adjoint extensions of an operator S which is symmetric in a Hilbert space H with deficiency indices 〈2,2〉 and which commutes with a non-trivial fundamental symmetry J of a Krein space (H,[⋅,⋅]),

SJ=JS.     

Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ) where Tⁿ is the n-dimensional torus and s?1. We prove that if P is s-globally hypoelliptic in Tⁿ then its transposed operator tP is s-globally solvable in Tⁿ, thus extending to the Gevrey classes the well-known analogous result in the corresponding C^∞ class.     

The similarity problem for J-nonnegative Sturm-Liouville operators

Sufficient conditions for the similarity of the operator with an indefinite weight r(x)=(sgnx)|r(x)| are obtained. These conditions are formulated in terms of Titchmarsh-Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x)=sgnx and , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q≡0, we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward-backward” diffusion equations.     

Two new q-exponential operator identities and their applications

In this paper, we use the q-Chu–Vandermonde formula to prove two new operator identities, which are the extensions of Liu's results. These two q-exponential operator identities are used to derive some q-summation formulas and q-integrals.     

Singular integral operators and norm ideals satisfying a quantitative variant of Kuroda's condition

A norm ideal C is said to satisfy condition (QK) if there exist constants 0<t<1 and 0<B<∞, such that ∥X^[k]∥_C?Bk^t∥X∥_C for every finite-rank operator X and every k∈N, where X^[k] denotes the direct sum of k copies of X. Let μ be a regular Borel measure whose support is contained in a unit cube Q in Rⁿ and let K_j be the singular integral operator on L²(Rⁿ,μ) with the kernel function (x_j-y_j)/|x-y|², 1?j?n. Let {Q_w:w∈W} be the usual dyadic decomposition of Q, i.e., {Q_w:|w|=?} is the dyadic partition of Q by cubes of the size 2^-?×?×2^-?. We show that if C satisfies (QK) and if ∥∑_w∈W2^|w|μ(Q_w)ξ_w⊗ξ_w∥_C^′<∞, where C^′ is the dual of C⁽⁰⁾ and {ξ_w:w∈W} is any orthonormal set, then K₁,…,K_n∈C^′. This is a very general obstruction result for the problem of simultaneous diagonalization of commuting tuples of self-adjoint operators modulo C.     

On a question of Brown, Douglas, and Fillmore

In this note we answer an old question of Brown, Douglas, and Fillmore [L. Brown, R.G. Douglas, P. Fillmore, Unitary equivalence modulo the compact operators and extensions of C^∗-algebras, in: Proc. Conf. Operator Theory, in: Lecture Notes in Math., vol. 345, Springer, Berlin, 1973, pp. 58-128].     

On k-hyperexpansive operators

This paper considers the k-hyperexpansive Hilbert space operators T (those satisfying , 1?n?k) and the k-expansive operators (those satisfying the above inequality merely for n=k). It is known that if T is k-hyperexpansive then so is any power of T; we prove the analogous result for T assumed merely k-expansive. Turning to weighted shift operators, we give a characterization of k-expansive weighted shifts, and produce examples showing the k-expansive classes are distinct. For a weighted shift W that is k-expansive for all k (that is, completely hyperexpansive) we obtain results for k-hyperexpansivity of back step extensions of W. In addition, we discuss the completely hyperexpansive completion problem which is parallel to Stampfli's subnormal completion problem.     

On the global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.     

On the Cohen strongly p-summing multilinear operators

In this paper, we introduce and study a new concept of summability in the category of multilinear operators, which is the Cohen strongly p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem and we compare the notion of p-dominated multilinear operators with this class by generalizing a theorem of Bu-Cohen.     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.     

Bounded linear non-absolutely summing operators

We show that, in certain situations, we have lineability in the set of bounded linear and non-absolutely summing operators. Examples on lineability of the set Πp(E,F)?Ip(E,F) are also presented and some open questions are proposed.     

On Eisenbud's and Wigner's R-matrix: A general approach

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.     

Gaps of operators via rank-one perturbations

In this note we consider rank-one perturbations of weighted shifts to examine distinctions between various sorts of weak hyponormalities, including p-hyponormality, p-paranormality, and absolute-p-paranormality. Our examples enable us to add to the small collection of examples that exhibit the gaps between these classes.     

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

     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

Composition operators from F(p, q, s) spaces to the nth weighted-type spaces on the unit disc

In this note we characterize the boundedness and compactness of the composition operator from the general function space F(p, q, s) to the nth weighted-type space on the unit disk, where the nth weighted-type space has been recently introduced by Stevo Stevi?.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

A note on generalized q-difference equations for q-beta and Andrews–Askey integral

Two q-difference equations with solutions expressed by q-exponential operator identities are investigated. As applications, two extensions of Ramanujan?s formulas for q-beta integral are given, two generalizations of Andrews–Askey integral are obtained. In addition, generating functions for generalized Al-Salam–Carlitz polynomials are deduced. At last, a generalized transformation identity is gained.     

On the rate of approximation by q modified Beta operators

In the present paper we propose the q analogue of the modified Beta operators. We apply q-derivatives to obtain the central moments of the discrete q-Beta operators. A direct result in terms of modulus of continuity for the q operators is also established. We have also used the properties of q integral to establish the recurrence formula for the moments of q analogue of the modified Beta operators. We also establish an asymptotic formula. In the end we have also present the modification of such q operators so as to have better estimate.