## Annals of Functional Analysis

### Cyclicity for Unbounded Multiplication Operators in $L^p$- and $C_0\,$-Spaces

#### Abstract

For every, possibly unbounded, multiplication operator in $L^p$-space, $p\in\,]0,\infty[$, on finite separable measure space we show that multicyclicity, multi-$*$-cyclicity, and multiplicity coincide. This result includes and generalizes Bram's much cited theorem from 1955 on bounded $*$-cyclic normal operators. It also includes as a core result cyclicity of the multiplication operator $M_z$ by the complex variable $z$ in $L^p(\mu)$ for every $\sigma$-finite Borel measure $\mu$ on $\mathbb{C}$. The concise proof is based in part on the result that the function $e^{-\left|z\right|^2}$ is a $*$-cyclic vector for $M_z$ in $C_0(\mathbb{C})$ and further in $L^p(\mu)$. We characterize topologically those locally compact sets $X\subset \mathbb{C}$, for which $M_z$ in $C_0(X)$ is cyclic.

#### Article information

Source
Ann. Funct. Anal., Volume 6, Number 2 (2015), 33-48.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.afa/1418997764

Digital Object Identifier
doi:10.15352/afa/06-2-4

Mathematical Reviews number (MathSciNet)
MR3292513

Zentralblatt MATH identifier
1312.47012

#### Citation

Zaigler, Sebastian; Castrigiano, Domenico P. L. Cyclicity for Unbounded Multiplication Operators in $L^p$- and $C_0\,$-Spaces. Ann. Funct. Anal. 6 (2015), no. 2, 33--48. doi:10.15352/afa/06-2-4. https://projecteuclid.org/euclid.afa/1418997764

#### References

• N.I. Achieser and I.M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum, Akademie-Verlag, Berlin 1968.
• I. Agricola and T. Friedrich, The Gaussian measure on algebraic varieties, Fund. Math. 159 (1999), no. 1, 91–98.
• E.A. Azoff and K.F. Clancey, Spectral multiplicity for direct integrals of normal operators, J. Operator Theory 3 (1980), no. 2, 213–235.
• J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94.
• L. Carleson, Mergelyan's theorem on uniform polynomial approximation, Math. Scand. 15 (1964) 167–175.
• D.P.L. Castrigiano and F. Hofmaier, Bounded point evaluations for orthogonal polynomials, Adv. Appl. Math. Sci. 10 (2011), no. 4, 373–392.
• D.P.L. Castrigiano and W. Roelcke, Topological Measures and Weighted Radon Measures, Alpha Science, Oxford 2008.
• K. Conrad, $L_p$-spaces for $0 < p < 1$, preprint.
• J.B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, 36. American Mathematical Society, Providence, RI, 1991.
• B. Fuglede, The multidimensional moment Problem, Expo. Math. 1 (1983), no. 1, 47–65.
• F. Hartogs and A. Rosenthal, Über Folgen analytischer Funktionen, Math. Ann. 100 (1928), no. 1, 212–263.
• E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York 1969.
• J.S. Howland, A decomposition of a measure space with respect to a multiplication operator, Proc. Amer. Math. Soc. 78 (1980), no. 2, 231–234.
• K.G. Kalb, Über die spektralen Vielfachheitsfunktionen des Multiplikationsoperators, Studia Math. 66 (1979), no. 1, 1–12.
• T.L. Kriete, An elementary approach to the multiplicity theory of multiplication operators, Rocky Mountain J. Math. 16 (1986), no. 1, 23–32.
• M.G. Nadkarni, Hellinger-Hahn type decompositions of the domain of a Borel function, Studia Math. 47 (1973) 51–62.
• B. Nagy, Multicyclicity of unbounded normal operators and polynomial approximation in $C$, J. Funct. Anal. 257 (2009), no. 6, 1655–1665.
• K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York 1967.
• V.A. Rohlin, On the fundamental ideas of measure theory, Mat. Sborn. 25 (1949) 107–150, Amer. Math. Soc. Transl. (1952), no. 71, 55 pp.
• H.A. Seid, Cyclic multiplication operators on $L_p$-spaces, Pacific J. Math. 51 (1974), no. 2, 549–562
• H.A. Seid, The decomposition of multiplication operators on $L_p$-Spaces, Pacific J. Math. 62 (1976), no. 1, 265–274.
• A. Shields, Cyclic vectors for multiplication operators, Michigan Math. J. 35 (1988), no. 3, 451–454.
• B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203.
• K. Stempak, J.L. Torrea, Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 (2003), no. 2, 443–472.
• E.L. Stout, Polynomial Convexity, Springer, New York 2007.
• F.H. Szafraniec, Normals, subnormals and an open question, Oper. Matrices 4 (2010), no. 4, 485–510.