ON THE SOLVABILITY OF CERTAIN (SSIE) WITH OPERATORS OF THE FORM B(r, s)

Given any sequence z = (zn)n≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n≥1 such that y/z = (yn/zn)n≥1 ∈ E; in particular, sz(c) denotes the set of all sequences y such that y/z converges. In this paper we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion each term of which is a sum or a sum of products of sets of sequences of the form Xa(T) and Xx(T) where a is a given sequence, the sequence x is the unknown, T is a given triangle, and Xa(T) and Xx(T) are the matrix domains of T in the set X . Here we determine the set of all positive sequences x for which the (SSIE) sx(c) (B(r, s)) sx(c)⊂ (B(r', s')) holds, where r, r', s' and s are real numbers, and B(r, s) is the generalized operator of the first difference defined by (B(r, s)y)n = ryn+syn−1 for all n ≥ 2 and (B(r, s)y)1 = ry1. We also determine the set of all positive sequences x for which ryn + syn−1 /xn → l implies r'yn + s'yn−1 /xn → l (n → ∞) for all y and for some scalar l. Finally, for a given sequence a, we consider the a–Tauberian problem which consists of determining the set of all x such that sx(c) (B(r, s)) ⊂ sa(c)

INFINITE MATRICES ASSOCIATED WITH POWER SERIES AND APPLICATION TO OPTIMIZATION AND MATRIX TRANSFORMATIONS

In this paper we first recall some properties of triangle Toeplitz matrices of the Banach algebra Sr associated with power series. Then for boolean Toeplitz matrices M we explicitly calculate the product MN that gives the number of ways with N arcs associated with M. We compute the matrix BN (i, j), where B (i, j) is an infinite matrix whose the nonzero entries are on the diagonals m &#8722; n = i or m &#8722; n = j. Next among other things we consider the infinite boolean matrix B+&#8734; that have infinitely many diagonals with nonzero entries and we explicitly calculate (B+&#8734;)N. Finally we give necessary and sufficient conditions for an infinite matrix M to map c (BN (i, 0)) to c.</p

INFINITE MATRICES ASSOCIATED WITH POWER SERIES AND APPLICATION TO OPTIMIZATION AND MATRIX TRANSFORMATIONS INFINITE MATRICES ASSOCIATED WITH POWER SERIES AND APPLICATION TO OPTIMIZATION AND MATRIX TRANSFORMATIONS INFINITE MATRICES ASSOCIATED WITH POWER SER

Abstract In this paper we first recall some properties of triangle Toeplitz matrices of the Banach algebra S r aociated with power series. Then for boolean Toeplitz matrices M we explicitly calculate the product M N that gives the number of ways with N arcs aociated with M. We compute the matrix B N (i, j), where B (i, j) is an infinite matrix whose the nonzero entries are on the diagonals m &amp;#x2212; n = i or m &amp;#x2212; n = j. Next among other things we consider the infinite boolean matrix B + ∞ that have infinitely many diagonals with nonzero entries and we explicitly calculate INFINITE MATRICES ASSOCIATED WITH POWER SERIES AND APPLICATION TO OPTIMIZATION AND MATRIX TRANSFORMATIONS Bruno de Malafosse and Adnan Yassine Abstract. In this paper we first recall some properties of triangle Toeplitz matrices of the Banach algebra S r associated with power series. Then for boolean Toeplitz matrices M we explicitly calculate the product M N that gives the number of ways with N arcs associated with M. We compute the matrix B N (i, j), where B (i, j) is an infinite matrix whose the nonzero entries are on the diagonals m − n = i or m − n = j. Next among other things we consider the infinite boolean matrix B + ∞ that have infinitely many diagonals with nonzero entries and we explicitly calculate`B + ∞´N . Finally we give necessary and sufficient conditions for an infinite matrix M to map c`B N (i, 0)´to c

On the Banach algebra ℬ(lp(α))

We give some properties of the Banach algebra of bounded operators ℬ(lp(α)) for 1≤p≤∞, where lp(α)=(1/α)−1∗lp. Then we deal with the continued fractions and give some properties of the operator Δh for h>0 or integer greater than or equal to one mapping lp(α) into itself for p≥1 real. These results extend, among other things, those concerning the Banach algebra Sα and some results on the continued fractions

On some BK spaces

We characterize the spaces sα(Δ), sα∘(Δ), and sα(c)(Δ) and we deal with some sets generalizing the well-known sets w0(λ), w∞(λ), w(λ), c0(λ), c∞(λ), and c(λ)

Calculations on some sequence spaces

We deal with space of sequences generalizing the well-known spaces w∞p(λ), c∞(λ,μ), replacing the operators C(λ) and Δ(μ) by their transposes. We get generalizations of results concerning the strong matrix domain of an infinite matrix A

Calculations in new sequence spaces

summary:In this paper we define new sequence spaces using the concepts of strong summability and boundedness of index $p>0$ of $r$-th order difference sequences. We establish sufficient conditions for these spaces to reduce to certain spaces of null and bounded sequences