The theory of frames for a Hilbert space plays a fundamental role in signal processing, image processing, data compression, sampling theory and more, as well as being a fruitful area of research in abstract mathematics. In this “tutorial” on abstract frame theory, we will try to point out the major directions of research in abstract frame theory and give some sample techniques from each of the areas. We will also bring out some of the important open questions, discuss some of the limitations of the existing theory, and point to some new directions for research.

In this report1, we present an order-theoretic version of the Cantor theorem. This result, which is based on the interplay of the notions of partial order and of completeness, permits to give a unified and simplified account to a long list of results related to the Bishop– Phelps theorem. We survey briefly only its simplest applications and refer the reader to [10] for a complete presentation of the results.

Let $R$ and $S$ be *-prime GPI rings with involution, with respective skew elements $K$ and $L$, with respective extended centroids $C$ and $D$, and let $\alpha :[K, K]/[K, K]\cap C\to [L, L]\cap D$ be a Lie isomorphism. If both involutions are of the second kind it is shown that $\alpha$ is determined by a related associative isomorphism and if the involutions are of different kinds it is shown that such a map $\alpha$ cannot exist (modulo some low-dimensional counterexamples).

A characterization of unconditional convergent series is given for the case of sequentially complete locally convex spaces. From it we show that if $E$ is a barrelled space with continuous dual $E'$, then ($E'$, $\beta (E'$, $E$)) contains no copy of ($c_0,~\|\cdot \|_\infty$) if and only if every continuous linear operator $T:E\to l_1$ is both compact and sequentially compact.

Let $C$ be a nonempty subset (not necessarily closed and convex) of a Hilbert space, and $S=\{T(t); t\geq 0\}$ be a semigroup of non-Lipschitzian mappings on $C$. In this paper we study almost-convergence of almost-orbits of $S$.

We introduce supsets and infsets for subsets of a partially ordered topological linear space. These notions generalize the usual notions of supremum and infimum in Riesz spaces. We shall investigate properties of supsets and infsets in this paper.

Sufficient conditions are given in order that a sequence of linear operators $L_n(\Lambda,\cdot)$ defined by $$ L_n(\Lambda,f):=\sum_{k=0}^n\lambda_{nk}\hat{f}(k)\chi_k\quad (n\in {N}_0),\quad \hat{f}(k):= \int_G\limits f\overline{\chi}_k\ (k\in N_0), $$ converges in $L^q$- norm to identity, where $f\in L^q(G)$, $q\in [1,\infty]$, $\lambda_{n0}=1\;(\forall n\in {N}_0)$, $\lambda_{nk}=0\;(\forall k\gt n,\forall n\in {N}_0)$ and $G$ is a general Vilenkin group. In case of bounded Vilenkin groups, our result coincides with an earlier result of Blyumin.

We will investigate the point spectrum on the imaginary axis of the linearized Boltzmann operator with an external-force potential in an unbounded domain. The boundary condition considered is the perfectly reflective boundary condition. We suppose that the boundary surface is sufficiently smooth, but is not cylindrical. In this case, the point spectrum on the imaginary axis is only equal to f0g.

We consider the exact boundary controllability for onedimensional linear heat equation with coefficients depending on the space variable and the time variable. We show that the functions of Gevrey class 2 are reachable when the initial functions are continuous.