Seminar: Berkay Kamil Taştan - Introduction to Aluthge transformation and its relation with the invariant subspace problem under the formation of an operator grid system (Mar 6, 2026)

Introduction to Aluthge transformation and its relation with the invariant subspace problem under the formation of an operator grid system

Berkay Kamil Taştan
Boğaziçi Üniversitesi

Let T∈B(H) be an operator with its polar decompostion T=U|T| such that ker(T)=ker(|T|)=ker(U). Then, "the Aluthge Transformation of T" is a non-linear transformation that can be formulated as Δ(T)=|T|12U|T|12.

The Invariant Subspace Problem'' is one of the major problems in functional analysis and operator theory searching whether for any given Banach space X and any given T∈B(X), there exists a non-trivial closed subspace S⊂X satisfying the condition T(S)⊂S.

There is a well known relation that H has a non-trivial invariant subspace for T∈B(H) if and only if for ΔT too. In this context, there exists a chain of operators ΔnT such that if H has a non-trivial invariant subspace for T, then for the rest of the operators in chain, but at this point, if for some n∈N and for some T∈B(H), Δn(T) is an operator for which H has a non-trivial invariant subspace, the information about the existence of invariant subspace for T can be reached. However, if there exists a quasi-affine relation throughout the chain and ∀n∈H, Δn(T) is quasi-affine, the use of a chain may not be enough. At this point, to be able to perform a comprehensive search, a grid system that has a shape similar to matrices and whose entries are Δn(Tm) has been constructed, and tried to model the verticle and horizontal motions on the grid. To be able to do that, we have delved into to the grid to decypher a so-called relation Δn(Tm)=Δm(Tn).

At first trials, we have considered T as being a centered operator which is an operator T such that the setO(T)={(U∗)j|T|Uj,|T|,Uk|T|(U∗)k:∀j,k∈N−{0}}commutes.