Recent Advances in Operator Theory and Operator Algebras

In this article the spectral theorem for right linear compact normal operators on quaternionic Hilbert spaces is proved. Though the version of spectral theorem for such operators in quaternionic Hilbert space is appeared in recent literature using the left multiplication and considering the Hilbert space to be slice complex linear, we present a different approach, which is similar to the classical setup. In our method we do not use the left multiplication and work with single operator throughout. It is observed that the whole spherical spectrum of a compact normal operator is determined by the standard eigenvalues and deduce that the spherical spectrum of any n n quaternion matrix has exactly ncomplex eigenvalues. We illustrate our method with an example and compare it with that of the method given by Ghiloni etal. Rashmirekha Patra. Sambalpur University, India. Title: A result on Nijenhuis Operator. Abstract: Construction of Nijenhuis Operator on generalized Tangent bundle (TM ⊕ T ∗M) of a Differentiable manifold M and trivial deformation on its maximally isotropic subspace using Nijenhuis Operator has been done in the light of Dorfmann’s work. Pawel Pietrzycki. Jagiellonian University, Poland. Title: The equality C∗2C2 = (C∗C)2 is not sufficient for quasinormality of a composition operator C in L-space Abstract: It is proved that a closed densely defined operator C is quasinormali if and only if the equality C∗nCn = (C∗C)n holds for n = 2, 3. Let W be bounded injective weighted shift which satisfies the equality W ∗nW n = (W ∗W ). We prove that operator W is then quasinormal. We will construct examples of bounded, non-quasinormal operator C which satisfies equality C∗nCn = (C∗C)n. An example of such a operator is given in the class of weighted shifts on directed trees. What is important, the directed tree used in the construction is rootless and therefore the operator in example is unitarily equivalent to a composition operator in L-space. Marek Ptak. University of Agriculture in Krakow, Poland. Title: C-symmetric operators and its preanihilator