In this article, we give an operator transform T(*) from class A operator to the class of hyponormal operators. It is different from the operator transform (T) over cap defined by M. Cho and T. Yamazaki. Then, we show that sigma(T) = sigma(T) over cap(*)) and sigma(alpha)((T) over cap)((*))/{0}, in case T belongs to class A. Next, we obtain some relations between (T) over cap and (T) over cap ((*))(*).

Let T = U|T| be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation \(\tilde T^{( * )} = |T^ * |^{\tfrac{1}{2}} U|T^ * |^{\tfrac{1}{2}} \) is called the Aluthge transformation and \(\tilde T_n \) means the n-th Aluthge transformation. Similarly, the transformation \(\tilde T^{( * )} = |T^ * |^{\tfrac{1}{2}} U|T^ * |^{\tfrac{1}{2}} \) is called the *-Aluthge transformation and \(\tilde T_n^{( * )} \) means the n-th *-Aluthge transformation. In this paper, firstly, we show that \(\tilde T^{( * )} = UV|\tilde T^{( * )} |\) is the polar decomposition of \(\tilde T^{( * )} \), where \(|T|^{\tfrac{1}{2}} |T^ * |^{\tfrac{1}{2}} = V||T|^{\tfrac{1}{2}} |T^ * |^{\tfrac{1}{2}} |\) is the polar decomposition. Secondly, we show that \(\tilde T^{( * )} = U|\tilde T^{( * )} |\) if and only if T is binormal, i.e., [|T|, |T*|]=0, where [A,B] = AB ‒ BA for any operator A and B. Lastly, we show that \(\tilde T_n^{( * )} \) is binormal for all non-negative integer n if and only if T is centered, and so on.

设H是一个复Hilbert空间,T是H上的一个有界线性算子,如果(Tx,x)≥0对一切x∈H成立,则称T是正算子,记为T≥0.

设H是复的Hilbert空间,H中的大写字母表示Hilbert空间中的有界线性算子.