Backward extensions of subnormal operators

Existence of Non-subnormal Polynomially Hyponormal Operators

In 1950, P. R. Halmos, motivated in part by the successful development of the theory of normal operators, introduced the notions of subnormality and hyponormality for (bounded) Hilbert space operators. An operator T is subnormal if it is the restriction of a normal operator to an invariant subspace; T is hyponormal if T*T > TT*. It is a simple matrix calculation to verify that subnormality impl...

Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded sub-normal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operator S lifts to the strong commutant of some tight selfadjoint extension of S. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreduci...

Hyperinvariant Subspaces for Some Subnormal Operators

In this article we employ a technique originated by Enflo in 1998 and later modified by the authors to study the hyperinvariant subspace problem for subnormal operators. We show that every “normalized” subnormal operator S such that either {(S∗nSn)1/n} does not converge in the SOT to the identity operator or {(SnS∗n)1/n} does not converge in the SOT to zero has a nontrivial hyperinvariant subsp...

One-step extensions of subnormal 2-variable weighted shifts

Observations on Primitive, Normal, and Subnormal Elements of Field Extensions

Let B1 and B2 be disjoint separable algebraic extensions of a field F, and let B = B1B2 be their composite. Let α1 be an element of B1 and α2 be an element of B2. Suppose α1 and α2 are primitive (resp. normal, resp. subnormal). We investigate the question of when α1 +α2 and α1α2 are necessarily primitive (resp. normal, resp. subnormal) elements of B. (A normal element of a Galois extension is d...