Subideals. A subideal of operators is an ideal of J (called a J-ideal) for J an arbitrary
ideal of B(H). Necessary and sufficient conditions are determined for a finitely generated
subideal to be also an ideal of B(H) and then these conditions are exploited to characterize
all finitely generated subideals. This generalizes to arbitrary ideals the 1983 work of Fong
and Radjavi who determined which principal (i.e., singly generated) ideals of the ideal of
compact operators are also ideals of B(H). Then a necessary and sufficient condition is found
for a countably generated subideal and a subideal generated by sets of cardinality strictly less
than the continuum to be also an ideal of B(H). This is based on the Hamel dimension of a
related quotient space. Then this condition is used to characterize these subideals and settle
additional general questions about subideals. The condition is a generalization of the notion
of soft-edged ideals discovered in 2007 by Kaftal and Weiss. Examples of subideals reveal
some striking differences between subideals and ideals of B(H). Also this work intersects
the study of subideals with the study of elementary operators with coefficient constraints.
Commutators of compact operators. The 1971 commutator problem asked by Pearcy
and Topping is investigated: Is every compact operator a single commutator of compact
operators? And a 1976 test question arising from work of Pearcy, Topping, Anderson and
Weiss: Are any strictly positive compact operators a single commutator of compact operators?
An affirmative answer to this test question for a whole class of strictly positive
compact operators is obtained.
Restricted diagonalization of compact normal operators. Every normal operator in F(H)
is diagonalizable by a unitary operator of the form 1 + A for 1 the identity operator and
A in F(H), as is elementary to show. But in an operator ideal I properly containing F(H) we found normal
compact operators that are not diagonalizable by unitary operators of the form 1 + A for
A in I. Indeed, a necessary condition that a normal operator X in I is diagonalizable by a
unitary operator of the form 1+A with A in I is: X-D in I^2 where D is a diagonal operator
unitarily equivalent to X. In addition, the spectral characterization of those A (compact or
not) for which 1 + A is a unitary operator is obtained, that is, 1 + A is unitary if and only
if A is a normal operator and its spectrum is contained in the circle -1 + T where T is the
unit circle. This group of unitary operators of the form 1 + A for A in I is used to obtain
uncountably many conjugacy classes of Cartan subalgebras of I.