ACTA SCIENTIARUM MATHEMATICARUM最新文献

T. M. Bisgaard [1] proved that the (*)-algebra (mathbb {C}[z,overline{z},z^{-1},overline{z}^{-1}])  has the moment property, that is, each positive linear functional on this (*)-algebra is a moment functional. We generalize this result to polynomials in d variables (z_1,dots ,z_d.) We prove that there exist (3d-2) linear polynomials as denominators such that the corresponding (*)-algebra has the moment property, while for 3 linear polynomials in case (d=2) the moment property always fails. Further, it is shown that for the real algebras (mathbb {R}[x,y,frac{1}{x^2+y^2}]) (the hermitean part of (mathbb {C}[z,overline{z},z^{-1},overline{z}^{-1}])) and (mathbb {R}[x,y,frac{x^2}{x^2+y^2},frac{xy}{x^2+y^2}]), all positive semidefinite elements are sums of squares. These results are used to prove that for the semigroup (*)-algebras of (mathbb {Z}^2), (mathbb {N}_0times mathbb {Z}) and ({textsf{N}}_+:={(k,n)in mathbb {Z}^2:k+nge 0}), all positive semidefinite elements are sums of hermitean squares.

In Part II, we deal with the “two-sided” Lawson order, which is the intersection of his orders (leqslant _l) and (leqslant _r), on U-semiabundant semigroups (presented as certain biunary semigroups). It is shown that, to a great extent, the order structure of these semigroups is determined by that of their set of projections. Our main topics of interest are existence of meets in such semigroups and rings and the possible lattice structure of their lower sections. In particular, every lower section of a U-semiabundant ring is shown, under certain simple assumptions, to be an orthomodular lattice, and explicit descriptions of the corresponding lattice operations and orthocomplementation are given.

This paper extends the study on weighted convolution operators presented in [J. Math. Anal. Appl., vol. 369, no. 2, pp. 712–718, 2010]. Specifically, we focus on the boundedness of a weighted Fourier convolution by one-dimensional Hermite functions via constructing some new (L_p)-norm estimates. An extended version of the Young theorem and a Hausdorff–Young type inequality are established. A sharp upper-bound coefficient for such inequalities is computed through Euler Gamma-function. Forward and reverse types of Saitoh inequality for the convolution proposed in this paper over weighted Lebesgue spaces are also formulated. The obtained results for the corresponding convolutions are then utilized to investigate the solvability of Fredholm integral equations and Cauchy-type problems as some applications. Solvability conditions and an explicit solution in (L_1) space are formulated. Finally, numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.

Using works of T. Ando and L. Gurvits, the well-known theorem of P.R. Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces is recast as a result for d-tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator d-tuples satisfying this matrix-positivity condition are called, herein, Toeplitz-contractive, and a characterisation of the Toeplitz-contractivity condition is presented. The matrix-positivity condition leads to definitions of new distance-measures in several variable operator theory, generalising the notions of norm, numerical radius, and spectral radius to d-tuples of operators (commuting, for the spectral radius) in what appears to be a novel, asymmetric way. Toeplitz contractive operators form a noncommutative convex set, and a scaling constant (c_d) for inclusions of the minimal and maximal matrix convex sets determined by a stretching of the unit circle (S^1) across d complex dimensions is shown to exist.

In this paper, we establish some new characterizations of operator monotone functions using matrix mean inequalities.

Recently the Karcher mean has been extended to the case of probability measures of positive operators on infinite-dimensional Hilbert spaces as the unique solution of a nonlinear operator equation on the convex Banach-Finsler manifold of positive operators. Let ((Omega ,mu )) be a probability space, and let (tau :Omega rightarrow Omega ) be a totally ergodic map. The main result of this paper is a new ergodic theorem for functions ( Fin L^1(Omega ,mathbb {P})), where (mathbb {P}) is the open cone of the strictly positive operators acting on a (separable) Hilbert space. In our result, we use inductive means to average the elements of the orbit, and we prove that almost surely these averages converge to the Karcher mean of the push-forward measure (F_*(mu )). From our result, we recover the strong law of large numbers and the “no dice” results proved by the third and fourth authors in the article Strong law of large numbers for the (L^1)-Karcher mean, Journal of Func. Anal. 279 (2020). From our main result, we also deduce an ergodic theorem for Markov chains with state space included in (mathbb {P}).

In this article, we conduct a comparative study of Crawford numbers and minimum moduli of bounded linear operators on Hilbert spaces. We provide a simple and distinct proof for the well-known equality of these two numbers for positive operators. Various estimates for Crawford numbers of operators have been derived, and the conditions under which an operator attains its Crawford number are discussed. We explore the relationships between subsets of the spectra and Crawford numbers of operators and also characterize the behaviour of Crawford numbers for self-adjoint operators. Additionally, we discuss the collection of operators whose Crawford numbers are zero, providing a detailed analysis and characterizing the conditions under which operators in this class attain their Crawford numbers.

Let (mathcal {M}_{n}) be the set of (ntimes n) complex matrices. Complete descriptions are given of the maps on (mathcal {M}_{n}) leaving invariant the numerical range of different kind of binary operations on matrices such as the skew product, the skew Lie product, the skew Jordan product and the skew semi-triple product, with no surjectivity assumption on them.

In this paper, we prove new numerical radius bounds that generalize some well-known results in the literature. For example, we prove that if A, B, X, Y are bounded linear operators on a complex separable Hilbert space H such that A and B are positive, then

This inequality generalizes a celebrated inequality proved by Kittaneh which states that:

.