We use elementary algebraic properties of left, right multiplication operators to prove some deep structural left $m$-invertible, $m$-isometric, $m$-selfadjoint and other related classes Banach space operators, often adding value extant results.

Limiting absorption principle is justified for second-order selfadjoint elliptic operators in two-dimensional spaces.

Some Jensen’s type inequalities for Log-Convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided. 2010 Mathematics Subject Classification: 47A63, 47A99

Some inequalities for positive linear maps of continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Applications for power function and logarithm are provided as well.