hardy space

In this paper,  we show that a shift operator on a separable Hilbert space with infinite multiplicity is strongly approximated by shift operators with finite multiplicities. Moreover, for an arbitrary shift operator $S$, we introduce the notion of an (asymptotic) $S$-Hankel operator and study its relation to the class of (asymptotic) $S$-Toeplitz operators.

In this paper, we study coposinormal composition operators and posinormal weighted composition operators on the Hardy space H2 (D). We show that if Wψ,ϕ is coposinormal on H2 (D), then ψ never vanishes on D also we prove that ϕ is univalent. Moreover, we study the commutant of a coposinormal weighted composition operator.

For a bounded analytic map ψ on the unit disk D and analytic self-map  φ of D, a weighted composition operator Cψ,φ on the Hardy space H2=H2(D) is defined by  Cψ,φf= ψ·f°φ. In this paper, we study the asymptotically Toeplitzness of weighted composition operators and their adjoints in different topology on B(H2).

Let For an analytic self-map of , Let be the composition operator with composite map  so that Let be a bounded analytic function on . The weighted composition operator is defined by . Suppose that is the Hardy space, consisting of all analytic functions defined on , whose Maclaurin cofficients are square summable.

The main aim of this paper is to construct an orthonormal wavelet on H2(D), the Hardy space of analytic functions on the open unit disc with square summable Taylor coefficients