In this paper, we discuss some properties of joint spectral {radius(jsr)} and  generalized spectral radius(gsr)  for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but  some are different. For example for a bounded set of scalar matrices,$Sigma$, $r_*...

Several representations of the generalized Drazin inverse of an anti-triangular block matrix in Banach algebra are given in terms of the generalized Banachiewicz--Schur form.  

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

several representations of the generalized drazin inverse of an anti-triangular block matrix in banach algebra are given in terms of the generalized banachiewicz--schur form.

Suppose that A and B are unital Banach algebras with units 1A and 1B , respectively, M is a unital Banach A−B-bimodule, T =   A M 0 B   is the triangular Banach algebra, X is a unital T -bimodule, XAA = 1AX1A, XBB = 1BX1B , XAB = 1AX1B and XBA = 1BX1A. Applying two nice long exact sequences related to A, B, T , X, XAA, XBB , XAB and XBA we establish some results on (co)homology of triangu...

Let A be a Banach algebra and M be a Banach left A-module. A linear map δ : M → M is called a generalized derivation if there exists a derivation d : A → A such that δ(ax) = aδ(x) + d(a)x (a ∈ A,x ∈ M). In this paper, we associate a triangular Banach algebra T to Banach A-module M and investigate the relation between generalized derivations on M and derivations on T . In particular, we prove th...