In 1950, P. R. Halmos, motivated in part by the successful development of the theory of normal operators, introduced the notions of subnormality and hyponormality for (bounded) Hilbert space operators. An operator T is subnormal if it is the restriction of a normal operator to an invariant subspace; T is hyponormal if T*T > TT*. It is a simple matrix calculation to verify that subnormality impl...

We are concerned with fractional powers of the so-called hyponormal operators of Putnam type. Under some suitable assumptions it is shown that if A, B are closed hyponormal linear operators of Putnam type acting on a complex Hilbert space H, then D((A+B)α) = D(Aα)∩D(Bα) = D((A+B)∗α) for each α ∈ (0,1). As an application, a large class of the Schrödinger’s operator with a complex potential Q ∈ L...

We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorith...

In this paper, we are concerned with conditions under which [p(T)]⁎=p‾(T⁎), where p(z) is a one-variable complex polynomial, and T an unbounded, densely defined, linear operator. Then, deal the validity of identities σ(AB)=σ(BA), A B two unbounded operators. The equations (TT⁎)⁎=TT⁎ (T⁎T)⁎=T⁎T, defined closable operator, also studied. particular interest will be paid to equation T⁎T=p(T) its va...

In this paper we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space H Cn of the unit circle. Firstly, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator. Secondly, we consider the gap...