Abstract:
We show that an absolute normalized norm on is strictly convex if and only if the corresponding convex function on is strictly convex. In this context the monotonicity property of these norms is discussed. We also introduce the notion of the direct sum of Banach spaces and equipped with the associated norm with and characterize the strict convexity of .

Abstract:
We study the connection between spectral properties of direct the sum of operators in the direct sum of Hilbert spaces and its coordinate operators.

Abstract:
In this work, we describe all normal extensions of a multipoint minimal operators generated by linear multipoint differential-operator expressions for first order in the Hilbert space of vector functions, in terms of boundary values at the endpoints of infinitely many separated subintervals. Also we investigate compactness properties of the inverses of such extensions.

Abstract:
Real interpolation spaces are used for solving some direct and inverse linear evolution problems in Banach spaces, on the ground of space regularity assumptions.

Abstract:
The von Neumann–Jordan (NJ-) constant for Lebesgue–Bochner spaces is determined under some conditions on a Banach space . In particular the NJ-constant for as well as (the space of -Schatten class operators) is determined. For a general Banach space we estimate the NJ-constant of , which may be regarded as a sharpened result of a previous one concerning the uniform non-squareness for . Similar estimates are given for Banach sequence spaces ( -sum of Banach spaces ), which gives a condition by NJ-constants of 's under which is uniformly non-square. A bi-product concerning 'Clarkson's inequality' for and is also given.

Abstract:
In this paper we discuss all normal extensions of a minimal operator generated by a linear multipoint differential-operator expression of first order in the Hilbert space of vector-functions on the finite interval in terms of boundary and interior point values. Later on, we investigate the structure of the spectrum, its discreteness and the asymptotic behavior of the eigenvalues at infinity for these extensions.

Abstract:
For an operator on a Banach space , let be the collection of all its invariant subspaces. We consider the index function on and we show, amongst others, that if is a bounded below operator and if , , then If in addition are index 1 invariant subspaces of , with nonzero intersection, we show that . Furthermore, using the index function, we provide an example where for some , holds .

Abstract:
In
this paper, the well known implicit function theorem was applied to study existence
and uniqueness of periodic solution of Duffing-type equation. Un-der
appropriate conditions around the origin, a unique periodic solution was
obtained.

Abstract:
In this paper, we introduce weakly compact version of the weakly countably determined (WCD) property, the strong WCD (SWCD) property. A Banach space X is said to be SWCD if there s a sequence (An) of weak ￠ — compact subsets of X ￠ — ￠ — such that if K ￠ X is weakly compact, there is an (nm) ￠ N such that K ￠ ￠ m=1 ￠ Anm ￠ X. In this case, (An) is called a strongly determining sequence for X. We show that SWCG ￠ ’SWCD and that the converse does not hold in general. In fact, X is a separable SWCD space if and only if (X, weak) is an ￠ μ0-space. Using c0 for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences.