Abstract:
An operator belongs to class for and if and only if if and only if where is generalized Aluthge transformation of , that is, . Class was introduced by Ito as a generalization of -hyponormality which was introduced by Aluthge and Wang. We shall show that if is a class operator for and , then belongs to class for all positive integer . As an immediate corollary of this result, we shall also show that if is a -hyponormal operator, then is also -hyponormal for all positive integer .

Abstract:
We study negative powers of Laguerre differential operators in $\R$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the {\sl convexity principle} to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\R$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

Abstract:
Recently, as a nice application of Furuta inequality, Aluthge and Wang (J. Inequal. Appl., 3 (1999), 279–284) showed that "if is a -hyponormal operator for , then is -hyponormal for any positive integer ," and Furuta and Yanagida (Scientiae Mathematicae, to appear) proved the more precise result on powers of -hyponormal operators for . In this paper, more generally, by using Furuta inequality repeatedly, we shall show that "if is a -hyponormal operator for , then is -hyponormal for any positive integer " and a generalization of the results by Furuta and Yanagida in (Scientiae Mathematicae, to appear) on powers of -hyponormal operators for .

Abstract:
Let $X\subset\mathbb{P}^{N}$ be an irreducible, non-degenerate variety. The generalized variety of sums of powers $VSP_H^X(h)$ of $X$ is the closure in the Hilbert scheme $Hilb_{h}(X)$ of the locus parametrizing collections of points $\{x_{1},...,x_{h}\}$ such that the $(h-1)$-plane $\left\langle x_{1},...,x_{h}\right\rangle$ passes trough a fixed general point $p\in\mathbb{P}^{N}$. When $X = V_{d}^{n}$ is a Veronese variety we recover the classical variety of sums of powers $VSP(F,h)$ parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of $VSP_H^X(h)$. In particular we will show how some birational properties, such as rationality, unirationality and rational connectedness, of $VSP_H^X(h)$ are inherited from the birational geometry of variety $X$ itself.

Abstract:
Hochster and Huneke proved in \cite{HH5} fine behaviors of symbolic powers of ideals in regular rings, using the theory of tight closure. In this paper, we use generalized test ideals, which are a characteristic $p$ analogue of multiplier ideals, to give a slight generalization of Hochster-Huneke's results.

Abstract:
Firstly, we will show the following extension of the results on powers of -hyponormal and -hyponormal operators: let and be positive integers, if is -hyponormal for , then: (i) in case , and hold, (ii) in case , and hold. Secondly, we will show an estimation on powers of -hyponormal operators for which implies the best possibility of our results. Lastly, we will show a parallel estimation on powers of -hyponormal operators as follows: let , then the following hold for each positive integer and : (i) there exists a log-hyponormal operator such that , (ii) there exists a -hyponormal operator such that .

Abstract:
The $\beta\gamma$ system is generalized by complex(rational) powers of the fields, which leads to a corresponding extension on the Fock space. Two different approaches to compute the Green functions of the physical operators are proposed. First the complex(rational) powers are defined via an integral representation,that allows to compute the conformal blocks, Green functions and structure constants of OPA. Next an approach based on a system of recursion equations for the Green functions is developed. A number of solutions of the system is found. A lot of possible applications is briefly discussed.

Abstract:
We are concerned with fractional powers of the so-called hyponormal operators of Putnam type. Under some suitable assumptions it is shown that if A, B are closed hyponormal linear operators of Putnam type acting on a complex Hilbert space ℍ, then D((A

Abstract:
we develop four identities concerning parameter differentiation of fractional powers of operators appearing in the tsallis ensembles of quantum statistical mechanics of nonextensive systems. in the appropriate limit these reduce to the corresponding differentiation identities of exponential operators of the gibbs ensembles of extensive systems derived by wilcox.

Abstract:
We develop four identities concerning parameter differentiation of fractional powers of operators appearing in the Tsallis ensembles of quantum statistical mechanics of nonextensive systems. In the appropriate limit these reduce to the corresponding differentiation identities of exponential operators of the Gibbs ensembles of extensive systems derived by Wilcox.