Operator Equations in Hilbert spaces

Abstract

The developmant of operator theory in Hilbert spaces, owes its origin to complex analysis. Thus, given a bounded linear operator A, we would like to view it as a generalised complex number viz:

A d: B .+ iC. L with Band C aS~3al and imaginary parts respectively.

However, it turnJ out that it is not always true that BC ~ CB as in the case of complex numbers. This in itself has been a drawback, which necessitated the I I i classification of operators. Hence, tne operator A for which BC ~ CB is said to be normal because it behaves like a c~opl8x number.

It 5s theref~rel quite natur~l t~at given an o~cratol equation like

.. , one would first address oneself to the problem of finding normal solutions. Simil~rly, we note that if A, Hand K are complex numbers a~d AH = KA, one would not require any ccndition on A in order to conclude that H ~ K. However, this is not always the case .if A, Hand K are bounded linear operators, It is there-

fore motivating enough to try an~ find conditions under which H ~ K. In the light of the remarks above, we address ourselves in this thesis t,o the following task

- ii -

jn the form of chapters:

[il On the operator equation AH = KA, in which we find sufficient conditions under which H = K and its consequences. (iil The operator equation AB + BA* = A*8 + BA = I,

in which we find necessary and sufficient conditions

for existenbe cf A or B.

'I-'"

(iiil We revist the operator equation AB + BA*

= A*B + BA = I, in which we apply some of the results of chapter one in order to find sufficient conditions for normality of A or B. It is infact shawn that some of the alr3ady published suffi- ciant conditions under which A is normal, can easily

be derived as corollaries tG our main results in

this chap t.er .

".,. tivl The operator equation TST*

.= S, in which we deduce

, unitary solutions.

Finally, while the study of each of these three operator equations may appear to be done in isolation, the three equations are shown infact to be interrelated. We would also like to note that the question of appli- cability of these operator equations is not of our primary concern in this thesis. Here, we concern ourselves with the abstract theory of these operator equations in which there is also sufficient merit.