Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S0 of S such that J is contained in the resolvent set of S0 and the associated Weyl function of the pair {S, S0} is monotone with respect to J , then for any self-adjoint operator R there exists ...

We establish the orthogonality of the range and the kernel of a normal derivation with respect to the unitarily invariant norms associated with norm ideals of operators. Related orthogonality results for certain nonnormal derivations are also given.

We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is pres...

The norm convergence of the Trotter{Kato product formula is established with ultimate optimal error bound for the self-adjoint semigroup generated by the operator sum of two self-adjoint operators. A generalization is also given to the operator sum of several self-adjoint operators.

Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂H a closed invariant subspace of A. Assuming that supspec(A0)≤ inf spec(A1), where A0 and A1 are restrictions of A onto the subspaces H0 and H1 = H ⊥ 0 , respectively, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V that are off-diagonal with respect to the decompositio...