Some Algebraic Operators and the Invariant Subspace Problem
Tóm tắt
We consider the resolvent algebra
$${R_A=\{T\in\mathcal{L} (X):\sup_{m \geq 0}\|(1+\,mA)T(1+\,mA)^{-1}\| < \infty \}}$$
, and Deddens’ algebra
$${B_A= \{T\in \mathcal{B}(H) : \sup_{n\geq 0}\|A^nTA^{-n}\|<\infty\}}$$
. It is shown that both R
A
and B
A–I
possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R
A
whenever R
A
≠ {A}′. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R
A
when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R
A
contains properly {A}′ whenever A has an eigenvalue other than zero.
Tài liệu tham khảo
Bercovici H.: Operator Theory and Arithmetic in H ∞. American Mathematical Society, Providence (1988)
Boas R.P.: Entire Functions. Academic Press, New York (1954)
Colojoara, I., Foias, C.: The Theory of Generalized Spectral Operators, Gordan and Breach, 1968
Deddens J.A., Wong T.K.: The commutant of analytic Toeplitz operators. Trans. Am. Math. Soc. 184, 261–273 (1973)
Deddens, J.A.: Another description of nest algebras in Hilbert spaces operators, Lecture notes in Mathematics No. 693, pp. 77–86. Springler, Berlin, 1978
Drissi D., Mbekhta M.: Operators with bounded conjugation orbits. Proc. Am. Math. Soc. 128, 2687–2692 (2000)
Drissi D., Mbekhta M.: Elements with generalized bounded conjugation orbits. Proc Am. Math. Soc. 129, 2011–2016 (2001)
Drissi D., Mbekhta M.: On the commutant and orbits of conjugation. Proc Am. Math. Soc. 134, 1099–1106 (2005)
Feintuch A., Markus A.: On operator algebras determined by a sequence of operator norms. J. Oper. Theory 60, 317–341 (2008)
Levin, B.Ja.: Distribution of Zeros of Entire Functions, Am. Math. Soc., Providence, 1964
Lomonosov V.: Invariant subspaces of family of operators that commute with a completely continuous operator. (Russian). Funkcional Anal. iPrilozhen 7(3), 55–56 (1973)
Nikolski N.: Treatise on the Shift Operator. Springer, Heidelberg (1986)
Pearcy C., Shields A.L.: A survey of the Lomonosov technique in the theory of invariant subspaces.. Am. Math. Soc. Surv. 13, 219–229 (1974)
Williams J.P.: On a boundedness condition for operators with singleton spectrum. Proc. Am. Math. Soc. 78, 30–32 (1980)