We prove a general result on the factorization of matrix-valued analytic functions. We deduce that if (E 0 , E 1 ) and (F 0 , F 1 ) are interpolation pairs with dense intersections, then under some conditions on the spaces E 0 , E 1 , F 0 and F 1 , we haveWe find also conditions on the spaces E 0 , E 1 , F 0 and F 1 , so that the following holdsSome applications of these results are also considered.
Introduction, notation and backgroundAll Banach spaces considered in this paper are complex. By an n-dimensional Banach space, we mean C n equipped with a norm.If X and Y are Banach spaces, then L(X, Y ), X ∨ ⊗Y and X ⊗Y denote, respectively, the Banach space of bounded operators from X into Y , The closure of X ⊗ Y in L(X * , Y ) equipped with the induced norm, and the completion of X⊗Y with respect to the projective tensor norm defined by :⊗Y and X ⊗Y are called respectively the injective and projective tensor product of X and Y . In the case when X and Y are both finite-dimensional, we have
In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy's integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.
Abstract. Using methods from classical analysis, sharp bounds for the ratio of differences of power means are obtained. Our results generalize and extend previous ones due to S. Wu (2005), and to S. Wu and L. Debnath (2011).Mathematics subject classification (2010): 26E60, 26D07.
In the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.
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