Advances in Operator Theory最新文献

This work focuses on the investigation of a quasilinear elliptic problem in the entire space (mathbb {R}^N), which involves the 1-Laplacian and 1-biharmonic operators, as well as potentials that can vanish at infinity. This research is conducted within the space of functions with bounded variation. The main result is proven using a version of the mountain pass theorem that does not require the Palais-Smale condition.

In this paper the boundary generating curves and numerical ranges of centrosymmetric matrices of orders up to 6 are characterized in terms of the matrices entries. These results extend previous ones concerning Kac-Sylvester matrices. The classification of all the possible boundary generating curves for centrosymmetric matrices of higher dimensions remains open.

This paper deals mainly with the idempotency of an operator or a matrix T given by (T=c_1 Pi _1 +c_2 Pi _2+cdots +c_nPi _n,) where n is an arbitrary positive integer, ({Pi _{1},Pi _{2},ldots ,Pi _{n}}) is a collection of mutually commutative idempotents, and (c_1,c_2,ldots ,c_n) are complex numbers. Some previous results in the cases of (n=2) and (n=3) are generalized, and meanwhile some new characterizations of the idempotency of T are obtained.

We study the integrability and S-factorizability of two-Lipschitz operators between pointed metric spaces and Banach spaces. We also introduce a factorization theorem and examine its connection with the bi-linearization maps. Finally, we establish a relationship between these two classes of operators.

The existence of a capacity solution to the strongly nonlinear degenerate problem, namely, (H(theta )+g(x,theta )=sigma (theta )|nabla psi |^{2}, {text {div}}(sigma (theta ) nabla psi )=0) in (Omega ) where (g(x,theta )) is a lower order term satisfies the sign condition but without any restriction on its growth and the operator H is of the form

is proved in the framework of Sobolev space of finite order.

Given an abstract semigroup S, by the Furstenberg fixed point property, we refer to a fixed point property of the following type: Whenever (S, X) is a nonexpansive compact flow with a certain property (P) and nonempty convex phase space X in a Hausdorff locally convex space (E, Q), then there exists a point (xin X) such that (s.x=x) for each (sin S). Motivated by the Furstenberg’s fixed point theorem on the existence of common fixed points for continuous affine compact flows, we are interested in investigating its natural nonlinear counterpart, and to this end we introduce and study a certain fixed point property for semigroups of nonexpansive mappings.

This paper presents spectral projection and modified projection methods for approximating the eigenelements of a compact integral operator with Green’s function-type kernels. The projection can either be the orthogonal projection or the interpolatory projection using Legendre polynomials. To the best of our knowledge, this paper is the first to consider the eigenvalue problem with Green’s kernels by global polynomials. We analyze the convergence of these methods and their iterated versions, and we establish superconvergence results. The effectiveness of the proposed approach is illustrated through various numerical tests.

Based on the Bonnabel–Sepulchre geometric mean for fixed rank matrices, we introduced paths of matrix means corresponding to the Kubo–Ando operator ones. We also observed that our mean includes the Batzies–Hüper–Machado–Silva Leite geometric means for fixed rank projection matrices. But these means are restricted to real ones and moreover it seems that it is not easy to extend them to operators on a complex (infinite dimensional) Hilbert space since these means were based on geometries for finite dimensional real spaces. In this paper, we introduce the general paths of means on a complex Hilbert space corresponding to those of the Kubo–Ando ones based on the infinite dimensional complex Grassmann geodesic in the sense of Andruchow.

The main objective of this paper is to use a new refinement of Young’s inequality to obtain two new scalar inequalities. As an application, we derive several new improvements of some well-known inequalities, which include the generalized mixed Schwarz inequality, numerical radius inequalities, Jensen inequalities and others. For example, for every (T,S in {mathcal {B(H)}}), (alpha in (0,1)) and (x, y in {mathcal {H}}), we prove that

where L is a positive 1-periodic function and r(S) is the spectral radius of S, which gives an improvement of the well-known generalized mixed Schwarz inequality:

where (|T| S=S^*|T|) and f, g are non-negative continuous functions defined on ([0, infty )) satisfying that (f(t) g(t)=t,(t ge 0)).

This paper deals with the existence and uniqueness of weak solution for a class of obstacle problem of the form

where (Im _{Lambda , h}) is a convex set defined below. By using the Young measure theory and Kinderlehrer and Stampacchia Theorem, we prove the existence and uniqueness result of the considered problem in the framework of generalized Sobolev space.