in this paper, we introduce the notion of an action $y_x$as a generalization of the notion of a module,and the notion of a norm $vt: y_xto f$, where $f$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: y_xto [0, 1]subseteq {bf r}$, with $bf r$  the set of all real numbers. a great many standard mappings on algebr...

Norm and essential norm of differences of differentiation composition operators between Bloch spaces have been estimated in this paper. As a result, we find characterizations for boundedness and compactness of these operators.

We present a proof that in Orlicz spaces the Amemiya norm and the Orlicz norm coincide for any Orlicz function ‘p. This gives the answer for an open problem. We also give a description of the Amemiya type for the Mazur-Orlicz F-norm.

In this note, we provide detailed characterization of operators in terms norm-attainability and norm estimates Banach algebras. particular, establish the necessary sufficient conditions for derivations also give their bounds norm-attainable classes.