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We introduce a new class of operators, which we will call the class of P -quasi- m -symmetric operators that includes m -symmetric operators and k -quasi m -symmetric operators. Some basic structural properties of this class of operators are established based on the operator matrix representation associated with such operators.

The tetrablock is the setThe closure of E is denoted by E. A tetra-inner function is an analytic map x from the unit disc D to E such that, for almost all points λ of the unit circle T, lim r↑1x(rλ) exists and lies in bE, where bE denotes the distinguished boundary of E. There is a natural notion of degree of a rational tetra-inner function x; it is simply the topological degree of the continuous map x| T from T to bE.In this paper we give a prescription for the construction of a general rational tetrainner function of degree n. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if x = (x 1 , x 2 , x 3 ) is a rational tetra-inner function of degree n, then x 1 x 2 − x 3 either is identically 0 or has precisely n zeros in the closed unit disc D, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x = (x 1 , x 2 , x 3 ) consists of the points in D for which x 1 x 2 − x 3 = 0 and the values of x at these points.

For a d-tuple of commuting operators S := (S1,..., Sd) ? B[X]d, m ? N and p ? (0,?), we define Q(p) m (S;u) := ? 0?k?m (-1)k (m k) (???Nd0 |?| = k k!/? ||S?u||p). As a natural extension of the concepts of (m,p)-expansive and (m,p)-contractive for tuple of commuting operators, we introduce and study the concepts of (m,?)-expansive tuple and (m,?)-contractive tuple of commuting operators acting on a Banach space. We say that S is (m,?)-expansive d-tuple (resp. (m,?)- contractive d-tuple) of operators if Q(p)m (S;u) ? 0 ? u ? X and p ? ? (resp. Q(p)m (S;u) ? 0 ? u ? X and p ? ?) . These concepts extend the definition of (m,?)-isometric tuple of bounded linear operators acting on Banach spaces was introduced and studied in [13].