Some estimates of norms of random matrices

We show that for any random matrix with independent mean zero entries

where is some universal constant.

     

Composition operators with maximal norm on weighted Bergman spaces

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces (in particular, on the space ) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space , where every inner function induces a composition operator with maximal norm.

     

A new proof of the two weight norm inequality for the one-sided fractional maximal operator

We give a new proof of the two weight norm inequality for the one-sided, fractional maximal operator, , simplifying the original proof of Martín-Reyes and de la Torre.

     

Semi-compactness of positive Dunford-Pettis operators on Banach lattices

We investigate Banach lattices on which each positive Dunford-Pettis operator is semi-compact and the converse. As an interesting consequence, we obtain Theorem 2.7 of Aliprantis-Burkinshaw and an element of Theorem 1 of Wickstead.

     

Essentially subnormal operators

An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form ``normal plus compact'.

The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam's Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

     

The essential selfcommutator of a subnormal operator

In this paper an easier proof is obtained of Alexandru Aleman's extension of an inequality of Axler and Shapiro for subnormal operators to the essential norm. The method is applied to show that a hyponormal operator whose essential spectrum has area zero must be essentially normal.

     

Compactness properties of operators dominated by AM-compact operators

We study several properties about the problem of domination in the class of positive AM-compact operators, and we obtain some interesting consequences on positive compact operators. Also, we give a sufficient condition under which a Banach lattice is discrete.

     

Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields

In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space.

We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat.

These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.

     

Boundedness of the Cesàro operator on mixed norm spaces

In this note, the boundedness of the Cesàro operator on mixed norm space , , is proved.

     

Elementary reverse Hölder type inequalities with application to operator interpolation theory

We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces.

     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

The index of triangular operator matrices

For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.

     

A characterization of bilateral operator weighted shifts being Cowen-Douglas operators

It is characterized when a bilateral operator weighted shift is a Cowen-Douglas operator.

     

On the range of the sum of monotone operators in general Banach spaces

The purpose of this paper is to generalize the Brézis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range is topologically almost equal to the sum where is a compatible topology in as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.

     

Applications of Wigner’s theorem to positive maps preserving norm of operator products

We apply Wigner’s theorem to positive maps on standard operator algebras that preserve norm of operator products or sum of singular values of operator products. It follows that such preservers are of the form ϕ(A) = U AU* with U either a unitary or antiunitary operator. Selected from Journal of Mathematical Study, 2004, 37(1): 11–16     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

Non-commutative operator Bohr inequality

It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|²?p|A|²+q|B|² with equality if and only if (1−p)A=B and γ||||A−B|²|||?|||p|A|²X+qX|B|²||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δ_A,B²(X)‖₂?‖p|A|²X+qX|B|²‖₂ with equality if and only if (1−p)AX=XB.     

A restriction theorem for Grushin operators

We study the Grushin operators acting on \(\mathbb{R}_x^{{d_1}} \times \mathbb{R}_t^{{d_2}}\) and defined by the formula \(L = - \sum\nolimits_{j = 1}^{{d_1}} {\partial _{{x_j}}^2} - {\sum\nolimits_{j = 1}^{{d_1}} {\left| {{x_j}} \right|} ^2}\sum\nolimits_{k = 1}^{{d_2}} {\partial _{{t_k}}^2} \). We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. M¨uller [Ann. of Math., 1990, 131: 567–587].