Complex Anal. Oper. Theory (2010) 4:245–255
DOI 10.1007/s11785-009-0022-4
Complex Analysis
and Operator Theory
Isometric Equivalence of Integration Operators
Nadia J. Gal · James E. Jamison ·
Aristomenis G. Siskakis
Received: 3 December 2007 / Accepted: 18 March 2009 / Published online: 18 April 2009
© Birkhäuser Verlag Basel/Switzerland 2009
Abstract We are interested in the isometric equivalence problem for the Cesàro
operator C ( f )(z ) =
1
z
z
0
f (ξ)
1
1−ξ
d ξ and an operator T
g
( f )(z ) =
1
z
z
0
f (ξ)g
(ξ)d ξ,
where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then
we generalize this to the isometric equivalence problem of two operators T
g
1
and T
g
2
on the Hardy space and Bergman space. We show that the operators T
g
1
and T
g
2
satisfy
T
g
1
U
1
= U
2
T
g
2
on H
p
, 1 ≤ p < ∞, p = 2 if and only if g
2
(z ) = λg
1
(e
i θ
z ), where
λ is a modulus one constant and U
i
, i = 1, 2 are surjective isometries of the Hardy
Space. This is analogous to the Campbell-Wright result on isometrically equivalence
of composition operators on the Hardy space.
Mathematics Subject Classification (2000) 47A05 · 47B37 · 47B49
Communicated by Daniel Alpay.
N. J. Gal (B )
Mathematics Department, University of Missouri,
323 Mathematical Sciences Bldg, Columbia, MO 65211, USA
e-mail: nadia@math.missouri.edu; galn@missouri.edu
J. E. Jamison
Department of Mathematical Sciences,
The University of Memphis, Memphis, TN 38152-3240, USA
e-mail: jjamison@memphis.edu
A. G. Siskakis
Department of Mathematics,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
e-mail: siskakis@math.auth.gr