Abstract
We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB p and the Dirichlet spaceD p. In the case ofB p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Banach,Theorie des Operations Lineares, Chelsea, Warzaw, 1932.
E. Berkson and A. Sourour,The hermitian operators on some Banach spaces, Studia Math.52 (1974), 628–634.
E. Berkson and H. Porta,Hermitian operators and groups of isometries in Hardy spaces, Trans. Amer. Math. Soc.185 (1973), 373–391.
E. Berkson and H. Porta,The group of isometries of H p, Ann. Mat. Pura Appl,(IV),CXIX (1979),231–238.
E. Berkson and H. Porta,The group of isometries on Hardy spaces of the n-ball and the polydisc, Glasgow Math. J.21 (1980) 199–204.
R.P. Boas,Isomorphism between H p and L p , Amer. J. of Math.77 (1955) 655–656.
M. Cambern,Isometries of certain Banach Algebras, Studia Math.,25 (1965), 217–225.
J. Cima and W. R. Wogen,On isometries of the Bloch spaces, Illinois J. Math.24 (1980), 313–316.
D. deLeeuw, W. Rudin, and J. Wermer,The isometries of some function spaces Proc. amer.Math.Soc.11 (1960), 694–698.
R. J. Fleming and J. E. Jamison,Isometries on certain Banach spaces, J. London Math. Soc.9 (1975), 121–127.
R.J. Fleming and J.E. Jamison,Hermitian operators and isometries on sums of Banach spaces, Proc. Edinburgh Math. Soc. (2)32 (1989), 169–191.
F.Forelli,The isometries of H p, Canad. J. Math.25 (1964), 721–728.
F. Forelli,A Theorem on isometries and its applications of it to isometries of H p(S) for 2<p<∞, Canad. J. Math.25 (1973), 284–289.
W. Hornor and J. E. Jamison,Isometrically equivalent composition operators Contemporary Math.213 (1998), 65–72.
N. J. Kalton and G. V. Wood,Orthonormal systems in Banach spaces and their applications, Math. Proc. Camb. Phil. Soc.79 (1976), 493–510.
N. J. Kalton and B. Randrianantoanina,Surjective isometries on rearrangement invariant spaces, Quart. J. Math. Oxford (2),45 (1994), 301–327.
C. J. Kolaski,Isometries of Bergman spaces over bounded Runge domains, Can. J. Math.33 (1981), 1157–1164.
C.J. Kolaski,Isometries of weighted Bergman spaces, Can. J. Math.34 (1982), 910–915.
C. J. Kolaski,Isometries of some smooth spaces of analytic functions, Complex Variables10 (1988), 115–122.
A. Koranyi and S. Vagi,Isometries of H p spaces of bounded symmetric domains, Canad. J. Math.28 (1976), 334–340.
S. G. Krantz and D. Ma,On isometric isomorphisms of the Bloch space in the unit ball, Michigan Math. J,36 (1989) 173–180
N. Lal and S. Merrill,Isometries of H p of the torus, Proc. Amer. Math. Soc.31 (1972), 465–471.
Chi-Kwong Li and B. Randrianantoanina,Isometries of direct sums of sequence spaces Preprint
J. Lindenstrauss and A. Pelczynski,Contributions to the theory of the classical Banach spaces J. Func. Anal.8 (1971) 225–249.
G. Lumer,Semi-inner-product spaces, Trans. Amer. Math. Soc.100 (1961), 29–43.
W.P. Novinger and D. M. Oberlin,Linear isometries of some space of analytic functions, Canad. J. Math.37 (1985), 62–76.
T. W. Palmer,Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc.133 (1968) 385–414.
J.R. Partington,Hermitian operators for absolute norms and direct sums, Linear Algebra Appl.23 (1979), 275–280.
R. Paya'-Albert,Numerical range of operators and structure of Banach spaces, Quart. J. Math. Oxford(2),33 (1982), 357–364.
V.V. Pellier,Hankel operators of class C p and their applications (rational approximation, Gaussian processes, the problem of majorizing operators), Math. USSR-Sbornik41 (1982), 443–479
A. Plotkin,Isometric Operators in L p spaces of analytic and harmonic functions, Zap. nauchn.semin. Leningr. Otd. Mat. Inst. Steklov.30, 130–145.
N. V. Rao and A. K. Roy,Linear isometries of some function spaces, Pacific J. Math.38 (1971), 177–192.
H. Rosenthal,Functional Hilbert sums, Pacific J. Math.124 (1986), 417–467.
W. Rudin,L p isometries and equi-measurability, Indiana Univ. Math. J.74 (1976), 228–232.
H. Schneider and R.E.L. Turner,Matrices hermitian for an absolute norm Linear and Multilinear Algebra1 (1973), 9–32
R. Schneider,Isometries of H p(U n), Canad. J. Math.25 (1973), 92–95.
A. Siskakis,Semigroups of Composition Operators Bull. Austral. Math. Soc.35 (1987), 397–406.
I. Vidav,Eine metrische Kennzeichnung der selbstadjungierten Operatoren, Math. Z66 (1956), 121–128.
Z. Wu and L. Yang,Multipliers Between Dirichlet Spaces, Integr. equ.oper. theory32 (1998), 482–492.
Kehe Zhu,Analytic Besov Spaces, J. Math. Anal.157 (1991), 318–336.