Contractions Satisfying the Absolute Value Property $$ |A|^2 \leq |A^2| $$

Abstract

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators A ∈ B(H) which satisfy the absolute value condition |A| 2 ≤ |A 2 |. It is proved that if A ∈ U is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator D = |A 2 |−|A| 2 is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in U , and it is shown that if normal subspaces of A ∈ U are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation δ A (X) = AX − XA with the commutant of A * is quasinilpotent.

ON k-QUASI-CLASS A CONTRACTIONS

“…Very recently B. P. Duggal and authors [3] extend these results to contractions in QA. In this paper, we extend these results to contractions in QA(k), which generalizes results proved for contractions in QA [2].…”

“…Recently B. p. Duggal, I. H. Jeon and C. S. Kubrusly [2] showed that if A is a class A contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the non-negative operator D = |A 2 | − |A| 2 is strongly stable (i.e., the power sequence {D n } converges strongly to 0). Very recently B. P. Duggal and authors [3] extend these results to contractions in QA.…”

“…All combinations are allowed, leading to the classes C 00 , C 01 , C 10 and C 11 of contractions [11,Page 72]. Duggal, Jeon and Kubrusly [2] showed that the following lemma; Lemma 2.3. If a class A contraction T has no nontrivial invariant subspace, then (a) T is a proper contraction and (b) the non-negative operator D = |T 2 | − |T | 2 is a strongly stable contraction (so that D ∈ C 00 ).…”

Contractions of Class Q and Invariant Subspaces

“…This was extended in [5] to contractions of class U (if a contraction T in U has no nontrivial invariant subspace, then both T and the nonnegative operator |T 2 | − |T | 2 are proper contractions), and to paranormal contractions in [6]: If a paranormal contraction T has no nontrivial invariant subspace, then T is a proper contraction and so is the nonnegative operator |T 2 | 2 − 2|T | 2 + I. In the present paper we extend this result to contractions of class Q.…”