The notion of a measure of noncompactness turns out to be a very important and useful tool in many branches of mathematical analysis. The current state of this theory and its applications are presented in the books [1,4,11] for example.The notion of a measure of weak noncompactness was introduced by De Blasi [8] and was subsequently used in numerous branches of functional analysis and the theory of differential and integral equations (cf. [2,3,9,10,11], for instance).In this note we summarize our...

Let X and Y be infinite dimensional Banach spaces and let L(X,Y) be the class of all (linear continuous) operators acting between X and Y. Mil'man [5] introduced the isometry spectrum I(T) of T ∈ L(X,Y) in the following way:I(T) = {α ≥ 0: ∀ ε > 0, ∃M ∈ S∞(X), ∀x ∈ SM, | ||Tx|| - α | < ε}},where S∞(X) is the set of all infinite dimensional closed subspaces of X and SM := {x ∈ M: ||x|| = 1} is the unit sphere of M ∈ S∞(X). (...)

Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the...

A construction method is presented for a class of simple C*-algebras whose basic properties -including their real ranks- can be computed relatively easily, using linear algebra. A numerival invariant attached to the construction determines wether a given algebra has real rank 0 or 1. Moreover, these algebras all have stable rank 1, and each nonzero hereditary sub-C*-algebra contains a nonzero projection, yet there are examples in which the linear span of the projections is not dense. (This phenomenon...

We introduce the spaces $M_{Y,\varphi }^1$, $M_{Y,\varphi }^{o,n}$, ${\tilde{M}}_{Y,\varphi }^o$ and $M_{Y,𝐝,\varphi }^o$ of multifunctions. We prove that the spaces $M_{Y,\varphi }^1$ and $M_{Y,𝐝,\varphi }^o$ are complete. Also, we get some convergence theorems.

In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II₁ factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ₀-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II₁ factor, equality of the induced traces implies unitary...

Let $H_p$ denote the usual Hardy space of analytic functions on the unit disc $(0\&lt;p\le \infty )$. We prove that for every function $f\in H_1$ there exists a linear operator $T$ defined on $L_1\left(\mathbf{T}\right)$ which is simultaneously bounded from $L_1\left(\mathbf{T}\right)$ to $H_1$ and from $L_{\infty }\left(\mathbf{T}\right)$ to $H_{\infty }$ such that $T\left(f\right)=f$. Consequently, we get the following results $(1\le p_0,p_1\le \infty )$:1) $(H_{p_0},H_{p_1})$ is a Calderon-Mitjagin couple;2) for any interpolation functor $F$, we have $F(H_{p_0},H_{p_1})=H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$, where$H\left(F\right(L_{p_0}\left(\mathbf{T}\right),L_{p_1}\left(\mathbf{T}\right)\left)\right)$ denotes the closed subspace of $F\left(L_{p_0}\right(\mathbf{T}),L_{p_1}(\mathbf{T}\left)\right)$ of all functions whose Fourier coefficients vanish on negative integers.These results also extend to Hardy...

Relations between different extensions of Toeplitz operators $T_{\phi }$ are studied. Additive properties of closed Toeplitz operators are investigated, in particular necessary and sufficient conditions are given and some applications in case of Toeplitz operators with polynomial symbols are indicated.