We consider dual Toeplitz operators on the orthogonal complements of the ock-Sobolev spaces of all nonnegative real orders. First, for symbols in a certain class containing all bounded functions, we study the problem of when an operator which is finite sums of the dual Toeplitz products is compact or zero. Next, for bounded symbols, we construct a symbol map and exhibit a short exact sequence associated with the $C^*$-algebra generated by all dual Toeplitz operators with bounded symbols.
Yong CHEN
,
Young Joo LEE
. SUMS OF DUAL TOEPLITZ PRODUCTS ON THE ORTHOGONAL COMPLEMENTS OF FOCK-SOBOLEV SPACES[J]. Acta mathematica scientia, Series B, 2024
, 44(3)
: 810
-822
.
DOI: 10.1007/s10473-024-0302-0
[1] Apostel C, Foias C, Voiculescu D. Some results on non-quasitriangular operators, IV. Rev Roumaine Math Pures Appl, 1973, 18: 487-513
[2] Benaissa L, Guediri H. Properties of dual Toeplitz operators with applications to Haplitz products on the Hardy space of the polydisk. Taiwanses J Math, 2015, 19: 31-49
[3] Chen Y, Lee Y J. Sums of dual Toeplitz products on the orthogonal complements of the Fock spaces. Integral Equations and Operator Theory, 2021, 93: Art 11
[4] Cho H, Choe B, Koo H. Fock-Sobolev spaces of fractional order. Potential Analysis, 2015, 43: 199-240
[5] Choe B, Koo H, Lee Y J. Sums of Toeplitz products with harmonic symbols. Revista Matematica Iberoamericana, 2008, 24: 43-70
[6] Guediri H. Dual Toeplitz operators on the sphere. Acta Mathematica Sinica, 2013, 29: 1791-1808
[7] Kong L, Lu Y. Some algebraic properties of dual Toeplitz operators. Houston J Math, 2018, 44: 169-185
[8] Lu Y. Commuting dual Toeplitz operators with pluriharmonic symbols. J Math Anal Appl, 2005, 302: 149-156
[9] Lu Y, Yang J. Commuting Dual Toeplitz operators on weighted Bergman spaces of the unit ball. Acta Mathematica Sinica, 2011, 27: 1725-1742
[10] Luecking D. Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk. J Funct Anal, 1992, 110: 247-271
[11] Rudin W.Function Theory in the Unit Ball of $\mathbb C^n$. New York: Springer-Verlag, 1980
[12] Stroethoff K, Zheng D.Algebraic and spectral properties of dual Toeplitz operators.
[13] Trans Amer Math Soc, 2002, 354: 2495-2520
[14] Yang Z, Zhang B, Lu Y. Dual Toeplitz operators on the ball. J Math Research with Applications, 2011, 41: 125-131
[15] Ye P, Yu T. Compactness of dual Toeplitz operators on the orthogonal complement of the Fock space on $\mathbb C^n$. Math Pract Theory, 2011, 41: 125-131
