| dc.description.abstract | The existence of direct, sum decompositions and factorizations of hounded linear operators acting on a Hilbert space appears to he one of the most difficult questions in the theory of linear operators. The direct sum decomposition problem is closely related to the invariant suhspace problem, which to date has very few affirmative answers regarding it. In this thesis we study the direct sum decomposition and factorization of some classes of operators in Hilbert spaces with a view to determining properties of the direct, summands of these operators, their invariant and hyperinvariant subspace lattices and factors for such operators.
This thesis is organized as follows: Chapter 1 is an introduction and is devoted largely to notations and terminology and examples of various concepts that we shall use in the rest of this t hesis.
Chapter 2 deals with the orthogonal direct sum decomposition of an arbitrary operator
into a normal and a completely non-normal part. In this chapter we show that a general operator T decomposes in this manner. We give conditions under which an operator has nontrivial normal and direct summands. We study this decomposition for operators in the same equivalence classes (quasisimilar, similar, unitarily equivalent, almost-similar operators). We give conditions under which a non-normal operator is normal.
Chapter 3 is on the direct sum decomposit ion of a contraction operator into a unitary and a completely non-unitary (c.n.u.) part. We give conditions under which a non- unitary operator is unitary. We show that a general operator enjoys this decomposition upon re-normalization (by dividing the operator by its norm). In so doing we show that t he problem of decomposing an operator into a normal and a c.n.n. part can be deduced from t he decomposition of a contraction operator. We pay special attention to the c.n.u parts of an operator and t he shift operators which play a very important role in t his kind of decomposition. We rise t he canonical backward shift as a model to aid t he decomposit ion of such operators. We also deduce the characteristic functions of some classes of operators and use t hem t o determine t he nat ure of t he original contraction
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operator.
In Chapter 4, we study t he invariant and hyperinvariant snhspaees for some classes of operators. Wo show that, these snhspaees reveal a lot. of information about the direct sum decompositions of linear operators. We investigate the topological structure of Lnl(T) and Hyjierlnt(T) for some operator classes containing T. We show that there is a one-to-one correspondence between the invariant lattice and the regular factorization of the characteristic function of a contraction operator T. We generalize this result to arbitrary operators.
Chapter 5 is on the factorization of some operators as a product of simpler operators (self-adjoint, unit ary, normal, project ions, idempotents, n-t h roots of t he ident ity. cyclic, scalar, etc.). We find necessary and sufficient conditions under which an operator can be expressed as a product of such simpler operators. We give necessary and sufficient condit ions on t he minimal number of such operator fact ors by improving on some known results.
By a canonical model of an operator we mean a natural representation of the operator in terms of simpler operators, and in a context in which more structure is present.
Most of the results in t he direct sum decompositions of an operator T will revolve around its nearness to being a normal operator ( [T*,T] = T*T — TT* = 0 ) and its nearness to being a unitary operator ( D^ = / — T*T = 0 or Df. = I — TT* = 0 ). | |
