Rend. Circ. Mat. Palermo (2014) 63:355–362
DOI 10.1007/s12215-014-0173-7
A Putnam–Fuglede commutativity theorem for class
A operators
B. P. Duggal · S. V. Djordjevi´ c · I. H. Jeon
Received: 6 May 2014 / Accepted: 3 October 2014 / Published online: 17 October 2014
© Springer-Verlag Italia 2014
Abstract Given a Hilbert space operator A ∈ B(H) with polar decomposition A = U | A|,
the class A(s, t ),0 < s , t ≤ 1, consists of operators A ∈ B(H) such that | A
∗
|
2t
≤
(| A
∗
|
t
| A|
2s
| A
∗
|
t
)
t
t +s
. Every class A(s, t ) operator is paranormal; prominent amongst the
subclasses of A(s, t ) operators are the class A
(
1
2
,
1
2
)
consisting of w-hyponormal operators
and the class A(1, 1) consisting of (semi-quasihyponormal [9, p. 93], or) class A operators.
This note considers Putnam–Fuglede type commutativity theorems for A(s, t ) operators to
prove that if the operators A, B
∗
∈ B(H) satisfy either of the conditions (1) 0 is a nor-
mal eigenvalue of both A, B
∗
∈ A(s, t ),0 < s, t ≤ 1; (2) 0 is a normal eigenvalue of
A ∈ A
(
1
2
,
1
2
)
and B
∗
is ( M-hyponormal or) dominant; (3) 0 is a normal eigenvalue of
A ∈ A(s, t ),0 < s, t ≤ 1, and both B
∗
and B
2
∗
are ( M-hyponormal or) dominant, then
(a) AX − XB = 0 ⇒ A
∗
X − XB
∗
= 0 for every X ∈ B(H), and (b) AXB − X = 0
implies A
∗
XB
∗
− X = 0 for every quasi-affinity X ∈ B(H).
Keywords Hilbert space · A(s, t ) operators · Commutativity theorem
Mathematics Subject Classification 47B20 · 47B47
B. P. Duggal (B)
8 Redwood Grove, Northfield Avenue, Ealing, London W5 4SZ, United Kingdom
e-mail: bpduggal@yahoo.co.uk
S. V. Djordjevi´ c
Benemérita Universidad Autónoma de Puebla, Rio Verde y Av. San Claudio, San Mauel, 72570 Puebla,
Puebla, Mexico
e-mail: slavdj@fcfm.buap.mx
I. H. Jeon
Department of Mathematics Education, Seoul National University of Education, Seoul 137-742, Korea
e-mail: jihmath@snue.ac.kr
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