Advances in Operator Theory最新文献

This work is devoted to the construction of a uniform asymptotics in the dimension of the matrix n tending to infinity of all eigenvalues in the case of a seven-diagonal Toeplitz matrix with a symbol having a zero of the sixth order, while the cases of symbols with zeros of the second and fourth orders were considered earlier. On the other hand, the results obtained refine the results of the classical work of Parter and Widom on the asymptotics of the extreme eigenvalues. We also note that the obtained formulas showed high computational efficiency both in sense of accuracy (already for relatively small values of n) and in sense of speed.

The purpose of this paper is to investigate the embedding theorems for Besov–Morrey spaces using the equivalence theorem for the K-functional and the modulus of continuity on Morrey spaces. First, we obtain some theorems in ball Banach function space and then focus on Morrey spaces. The Marchaud’s inequality on Morrey spaces and a specific case of embedding theorems for Sobolev–Morrey spaces are crucial tools. We show that the Besov–Morrey space (B_{alpha , a}^{p,lambda }(mathbb {R}^{n})) is continuously embedded in the Morrey-Lorentz space (mathcal {M}_{q,p}^{lambda }(mathbb {R}^{n})), and also, for any (alpha , beta > 0) and (1< ale p < q le infty ), the Besov–Morrey space (B_{alpha + beta , a}^{p,lambda }(mathbb {R}^{n})) is continuously embedded in the Besov–Morrey space (B_{beta , a}^{q,lambda }(mathbb {R}^{n})).

Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation constructions in operator theory, and the second to general classes of stochastic processes. For the latter, we apply our operator valued kernel-results in order to build new Hilbert space-valued Gaussian processes, and to analyze their structures of covariance configurations.

The theory of (p, r, s)-summing and (p, r, s)-nuclear linear operators on Banach spaces was developed by Pietsch in his book on operator ideals (Pietsch in Operator ideals, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980, Chapters 17 and 18) Due to recent advances in the theory of ideals of Bloch maps, we extend these concepts to Bloch maps from the complex open unit disc (mathbb {D}) into a complex Banach space X. Variants for (r, s)-dominated Bloch maps of classical Pietsch’s domination and Kwapień’s factorization theorems of (r, s)-dominated linear operators are presented. We define analogues of Lapresté’s tensor norms on the space of X-valued Bloch molecules on (mathbb {D}) to address the duality of the spaces of ((p^*,r,s))-summing Bloch maps from (mathbb {D}) into (X^*). The class of (p, r, s)-nuclear Bloch maps is introduced and analysed to give examples of (p, r, s)-summing Bloch maps.

The invertibility of Toeplitz plus Hankel operators (T(mathcal {A})+H(mathcal {B})), (mathcal {A},mathcal {B}in L^infty _{dtimes d}(mathbb {T})) acting on vector Hardy spaces (H^p_d(mathbb {T})), (1<p<infty ), is studied. Assuming that the generating matrix functions (mathcal {A}) and (mathcal {B}) satisfy the equation

where (widetilde{mathcal {A}}(t):=mathcal {A}(1/t)), (widetilde{mathcal {B}}(t):=mathcal {B}(1/t)), (tin mathbb {T}), we establish sufficient conditions for the one-sided invertibility and invertibility of the operators mentioned and construct the corresponding inverses. If (d=1), the above equation reduces to the known matching condition, widely used in the study of Toeplitz plus Hankel operators with scalar generating functions.

We prove some splitting results for bilinear summing operators and as a consequence Pietsch type composition results. Some examples are given.

In this paper, we prove several interpolating inequalities for unitarily invariant norms of matrices. Using the log-convexity of certain functions, enables us to obtain refinements of recent norm inequalities. Generalizations of some well-known norm inequalities are also given.

In 1996, X. Li and D. Yang found the best possible range of index (alpha ) for the boundedness of some sublinear operators on Herz spaces ({dot{K}}_q^{alpha , p}({{mathbb {R}}}^n)) or (K_q^{alpha , p}({{mathbb {R}}}^n)), under a certain size condition. Also, in 1994 and 1995, S. Lu and F. Soria showed that concerning the boundedness of above sublinear operator T on ({dot{K}}_q^{alpha , p}({{mathbb {R}}}^n)) or (K_q^{alpha , p}({{mathbb {R}}}^n)) with critical index of (alpha ), T is bounded on the power-weighted Herz spaces ({dot{K}}_q^{alpha , p}(w)({{mathbb {R}}}^n)) or (K_q^{alpha , p}(w)({{mathbb {R}}}^n)). In this paper, we will prove that for the two-power-weighted Herz spaces ({dot{K}}_{q_1}^{alpha , p}(w_1,w_2)({{mathbb {R}}}^n)) or (K_{q_2}^{alpha , p}(w_1,w_2)({{mathbb {R}}}^n)) with indices beyond critical index of (alpha ), the above T is bounded on them. Further, we will extend this result to a sublinear operator satisfying another size condition and a pair of Herz spaces (K_q^{alpha , p}(w_{beta _1},w_{beta _2})({{mathbb {R}}}^n)) and (K_q^{alpha , p}(w_{gamma _1},w_{gamma _2})({{mathbb {R}}}^n)). Moreover, we will also show the result of weak version of the above boundedness.

In this paper, we study the commutators of the Hardy operators on the Heisenberg group. We get some sufficient and necessary conditions for the compactness of the commutators of the Hardy operators on the Heisenberg group.

The (exp _q(z)) function is the standard q-analogue of the exponential. Since not much is known about this function, our aim is to give a contribution to the knowledge on (exp _q). After proving some simpler but new relations for it, we make a complete description of the inverse map of (exp _q(z)), including its branch structure and Riemann surface.