On direct integral of operators (Q2827650)

In the first main part of this paper (Section 2), the authors give two theorems concerning the connection between the spectrum and resolvent sets of the direct integral of operators and its component operators. Some formulas concerning this link are given in a corollary. Also, it is remarked that Theorem 2.1 (ii) is not true in the general case. On the other hand, some examples are provided satisfying the theorems. In the second main part (Section 3), four theorems concerning the relationship between power and polynomially boundedness of direct integral operators in the direct integral of Hilbert spaces and its component operators are presented. It is important to note that an example, which shows that the converse of Theorem 3.1 is not true, is provided. This paper contains remarkable theorems for this area.