Let T be a w-hyponormal operator on a Hilbert space H, T its Aluthge transform, λ an isolated point of the spectrum of T, and E λ and E λ the Riesz idempotents, with respect to λ, of T and T , respectively. It is shown that (2000). Primary 47A10, 47A53, 47B20. Mathematics Subject Classification

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the L ∞ -bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose L ∞ -norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools.

In this paper we show some structural properties of a p-quasihyponormal operator via Löwner-Heinz inequality and Hansen inequality. As important applications, it is shown that the normal parts of quasisimilar injective p-quasihyponormal operators are unitarily equivalent and quasisimilar injective p-quasihyponormal operators have same spectra and essential spectra.

This paper is devoted to study the several existence results of a sequence of infinitely many solutions to the nonlocal elliptic problem involving the fractional p(x)-Laplacian without assuming the Ambrosetti and Rabinowitz type condition. The strategy of the proof for these results is to approach the problem variationally by using the fountain theorem and the dual fountain theorem. In addition, we prove that the sequence of weak solutions becomes bounded solutions.

In this paper, we show the norm attaining paranormal operators have a nontrivial invariant subspace. Also, we show the norm attaining quadratically hyponormal weighted shift is subnormal.