Abstract
We develop techniques to study bounded linear transformations T such that \({\sum_{m,n} c_{m,n} T^{*n}T^m = 0}\) where \({c_{m,n} \in {\bf C}}\) and all but finitely many c m,n are zero. Such an operator is called a hereditary root of the polynomial \({p(x,y) = \sum_{m,n} c_{m,n} y^{n}x^{m}}\) . Self-adjoint operators, isometries, n-symmetric operators and m-isometries are all examples of roots of polynomials. The techniques developed include finding maximal invariant subspaces satisfying certain operator-theoretic properties, descriptions of the spectral picture of a root, resolvent inequalities for a root, and additional properties which can be derived when knowing more about the spectrum of a particular root. The results involving maximal invariant subspaces are applicable to operators which are not roots of polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agler, J.: Subjordan operators, Thesis. Indiana University (1980)
Agler, J.: An abstract approach to model theory. Survey of some recent results in operator theory, vol. II. Pitman Research Notes in Mathematics Series, No. 192
Agler J.: A disconjugacy theorem for Toeplitz operators. Am. J. Math. 112, 1–14 (1980)
Agler J.: Hypercontractions and subnormality. J. Oper. Theory 13, 203–217 (1985)
Agler J.: The Arveson extension theorem and coanalytic models. J. Integr. Equ. Oper. Theory 5, 608–631 (1982)
Agler J.: Subjordan operators: Bishop’s theorem, spectral inclusion, and spectral sets. J. Oper. Theory 7, 373–395 (1982)
Agler J.: Rational dilation of an annulus. Ann. Math. 121, 537–563 (1985)
Agler J., Helton W.J., Stankus M.: Classification of hereditary matrices. Linear Algebra Appl. 274, 125–160 (1998)
Agler J., Stankus M.: m-isometric transformations on Hilbert Space I. J. Integr. Equ. Oper. Theory 21(4), 383–429 (1995)
Agler J., Stankus M.: m-isometric transformations on Hilbert space II. J. Integr. Equ. Oper. Theory 23(1), 1–48 (1995)
Agler J., Stankus M.: m-isometric linear transformations on Hilbert space III. J. Integr. Equ. Oper. Theory 24(4), 379–421 (1996)
Berberian S.K.: Approximate proper values. Proc. Am. Math. Soc. 13, 111–114 (1962)
Conway J.B.: A Course in Functional Analysis. Springer, New York (1985)
Curto R.E., Putinar M.: Existence of non-subnormal polynomially hyponormal operators. B.A.M.S. 25(2), 373–378 (1991)
Helton, J.W.: Operators with a representation as multiplication by x on a Sobolev space. Colloquia Math. Soc. Janos Bolyai 5, Hilbert Space Operators, Tihany, pp. 279–287 (1970)
McCullough, S.A.: 3-Isometries, Thesis. University of California, San Diego (1987)
McCullough S.A.: Sub Brownian operators. J. Oper. Theory 22(2), 291–305 (1989)
Richter S.: Invariant subspaces of the Dirichlet shift. J. Reine Agnew. Math. 386, 205–220 (1988)
Radjavi H., Rosenthal P.: On roots of normal operators. J. Math. Anal. Appl. 34, 653–664 (1971)
Radjavi H., Rosenthal P.: Invariant Subspaces. Springer, New York (1973)
Stankus, M.: Isosymmetric Linear Transformations on Complex Hilbert Space, Thesis (Ph.D.). University of California, San Diego, pp. 1–80 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stankus, M. m-Isometries, n-Symmetries and Other Linear Transformations Which are Hereditary Roots. Integr. Equ. Oper. Theory 75, 301–321 (2013). https://doi.org/10.1007/s00020-012-2026-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-012-2026-0