Extensions of Hilbert Modules and Hankel Operators

Abstract

Extensions of the Hardy and the Bergman modules over the disc algebra are studied. The author relates extensions of these canonical modules to the symbol spaces of corresponding Hankel operators. In the context of function theory, an explicit formula of Ext(L _α ²(D), H ²(D)) is obtained. Finally, it is also proved that Ext(L _α ²(D), L _α ²(D)) ≠ 0. This may be the essential difference between the Hardy and the Bergman modules over the disk algebra.

Keywords

Hilbert module / Hankel operator / Disc algebra / Symbol space / 47B35 / O177.1 / O177.6

References

[1] Chen, X. M. & Guo, K. Y., Cohomology and extensions of hypo-Silov modules over unit modulus algebras (to appear in J. Operator Theory).

[2] Guo K Y. Algebraic reduction for Hardy submodules over polydisk algebras. J. Operator Theory, 1999, 41: 127-138

[3] Guo K Y. Characteristic spaces and rigidity for analytic Hilbert modules. J. Funct. Anal., 1999, 163: 133-151

[4] Guo K Y. Normal Hilbert modules over the ball algebras. Studia Mathematica, 1999, 135: 1-12

[5] Guo K Y, Chen X M. Extensions of Hardy modules over the polydisk algebra. Chin. Ann. of Math., 1999, 20B(1): 103-110

[6] Carlson J F, Clark D N. Cohomology and extensions of Hilbert modules. J. Funct. Anal., 1995, 128: 278-306

[7] Carlson J F, Clark D N, Foias C, Williams J P. Projective Hilbert A(D)-modules. New York J. Math., 1994, 1: 26-38

[8] Pisier G. A polynomially bounded operator on Hilbert space which is not similar to a contraction. J. Amer. Math. Soc., 1997, 10: 351-369

[9] Douglas R G, Paulsen V I. Hilbert modules over function algebras, 1989, New York: Longman Scientific & Technical

[10] Duren, P. L., Theory of H ^p space, Academic Press, Inc., New York, 1970.

[11] McDonald G, Sundberg C. Toeplitz operators on the disc. Indiana University Math. J., 1979, 28: 595-611

[12] Hastings W W. A Carleson measure theorem for Bergman spaces. Proc. Amer. Math. Soc., 1975, 52: 237-241

[13] Axler S. The Bergman space, the Block space, and commutators of multiplication operators. Duke Math. J., 1986, 53: 315-332

[14] Carlson J F, Clark D N. Projectivity and extensions of Hilbert modules over A(D ⁿ). Michigan Math. J., 1997, 44: 365-373

[15] Ferguson S H. Backward shift invariant operator ranges. J. Funct. Anal., 1997, 150: 526-543