The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

An operator inequality related to Jensen's inequality

For bounded non-negative operators and , Furuta showed

We will extend this as follows: implies

where is a harmonic mean of and . The idea of the proof comes from Jensen's inequality for an operator convex function by Hansen-Pedersen.

     

The Furuta inequality with negative powers

Let be bounded linear operators on a Hilbert space satisfying . Furuta showed the operator inequality
as long as positive real numbers satisfy and . In this paper, we show this inequality is valid if negative real numbers satisfy a certain condition. Also, we investigate the optimality of that condition.

     

Complete form of Furuta inequality

Let and be bounded linear operators on a Hilbert space satisfying . The well-known Furuta inequality is given as follows: Let and ; then . In order to give a self-contained proof of it, Furuta (1989) proved that if , and , then .

This paper aims to show a sharpening of Furuta (1989): Let , and ; then . We call it the complete form of Furuta inequality because the case of it implies the essential part () of Furuta inequality for by the famous Löwner-Heinz inequality. Afterwards, the optimality of the outer exponent of the complete form is considered. Lastly, we give some applications of the complete form to Aluthge transformation.

     

Abstract blowing down

Assume that is a surface over an algebraically closed field . Let be obtained from by blowing up a smooth point and let be the exceptional curve. Let be the category of coherent sheaves on . In this note we show how to recover from , if we know the object .

     

Gradient estimates for positive solutions of the Laplacian with drift

Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where If is a vector field such that and on for some nonnegative constants and then we show that any positive solution of the equation satisfies the estimate

on , for all In particular, for the case when this estimate is advantageous for small values of and when it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).

     

Operators with bounded conjugation orbits

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

     

A decreasing operator function associated with the Furuta inequality

Let with and let and . As a generalization of a result due to Furuta, it is shown that the operator function

is decreasing for and if . Moreover, if and , then is decreasing for and . The latter result is an extension of an earlier result of Furuta.

     

On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality

where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,

where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:

and the corresponding eigenfunction spaces are

     

The first occurrence for the irreducible modules of general linear groups in the polynomial algebra

Let be a prime number and let be the group of all invertible matrices over the prime field . It is known that every irreducible -module can occur as a submodule of , the polynomial algebra with variables over . Given an irreducible -module , the purpose of this paper is to find out the first value of the degree of which occurs as a submodule of , the subset of consisting of homogeneous polynomials of degree . This generalizes Schwartz-Tri's result to the case of any prime .

     

On Korenblum's maximum principle

Let be the Bergman space over the open unit disk in the complex plane. Korenblum's maximum principle states that there is an absolute constant , such that whenever ( ) in the annulus , then . In this paper we prove that Korenblum's maximum principle holds with .

     

Splittings of Banach spaces induced by Clifford algebras

Let be an infinite-dimensional Hilbert space of density character . By representing as a module over an appropriate Clifford algebra, it is proved that possesses a family of proper closed nonzero subspaces such that

Analogous results are proved for spaces and for and () when is an arbitrary nonzero Banach space.

     

Relations between the Taylor spectrum and the Xia spectrum

Let be a doubly commuting -tuple of -hyponormal operators with unitary operators from the polar decompositions . Let and . In this paper, we will show relations between the Taylor spectrum and the Xia spectrum .

     

Carleson measures and some classes of meromorphic functions

For let be the Möbius transformation defined by , and let be the Green's function of the unit disk . We construct an analytic function belonging to for all , , but not belonging to meromorphic in and for any , . This gives a clear difference as compared to the analytic case where the corresponding function spaces ( and ) are same.

     

Bloch radius, normal families and quasiregular mappings

Bloch's Theorem is extended to -quasiregular maps , where is the standard -dimensional sphere. An example shows that Bloch's constant actually depends on for .

     

Trudinger type inequalities in and their best exponents

We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

     

Every -regular

We prove the following: Theorem A. If is a -regular ultrafilter, then either

(a)

is -regular, or

(b)

the cofinality of the linear order is , and is -regular for all .

Corollary B. Suppose that is singular, and either is regular, or . Then every -regular ultrafilter is -regular.

We also discuss some consequences and variations.

     

On an elementary approach to the fractional Hardy inequality

Let be the usual Hardy operator, i.e., . We prove that the operator is bounded and has a bounded inverse on the weighted spaces for and . Moreover, by using these inequalities we derive a somewhat generalized form of some well-known fractional Hardy type inequalities and also of a result due to Bennett-DeVore-Sharpley, where the usual Lorentz norm is replaced by an equivalent expression. Examples show that the restrictions in the theorems are essential.

     

A note on commutativity up to a factor of bounded operators

In this note, we explore commutativity up to a factor for bounded operators and in a complex Hilbert space. Conditions on possible values of the factor are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation and explore the structures of and that satisfy for some A quantum effect is an operator on a complex Hilbert space that satisfies The sequential product of quantum effects and is defined by We also obtain properties of the sequential product.

     

A Fatou theorem for the equation

In one space dimension and for a given function (say such that in some interval), the equation can be thought of as describing the energy per unit volume in a Stefan-type problem where the latent heat of the phase change is given by . Given a solution to this equation, we prove that for a.e. , there exists where is the Radon-Nikodym derivative of the initial trace with respect to Lebesgue measure and are the parabolic ``non-tangential" approach regions. Since only is continuous, while is usually not, does not hold in general.