Compression of Slant Toeplitz Operators on the Hardy Space of n -Dimensional Torus
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Czechoslovak Mathematical Journal
22 pp
Online first
COMPRESSION OF SLANT TOEPLITZ OPERATORS ON THE HARDY SPACE OF n-DIMENSIONAL TORUS Gopal Datt, Shesh Kumar Pandey, Delhi Received February 27, 2019. Published online July 21, 2020.
Abstract. This paper studies the compression of a kth-order slant Toeplitz operator on the Hardy space H 2 (Tn ) for integers k > 2 and n > 1. It also provides a characterization of the compression of a kth-order slant Toeplitz operator on H 2 (Tn ). Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of kth-order slant Toeplitz operator on the Hardy space H 2 (Tn ) of n-dimensional torus Tn . Keywords: Toeplitz operator; compression of slant Toeplitz operator; n-dimensional torus; Hardy space MSC 2020 : 47B35
1. Introduction Throughout the paper, the set of all complex numbers, the open unit disc and the unit circle in the complex plane are denoted by C, D and T, respectively. The theory of slant Toeplitz operators on L2 (T) was developed by Ho (see [5], [7]), who investigated several features of the slant Toeplitz operators on L2 (T), such as norms, spectrum and eigen spaces etc. Arora and Batra in [1] and [2] extended this concept to the kth-order slant Toeplitz operators on L2 (T) and its compression on H 2 (T). Ding, Sun and Zheng studied Toeplitz operators and their commutativity on the bi-disk in [4]. Lu and Zhang discussed the notion of commuting Hankel and Toeplitz operators on the Hardy space of the bi-disk, see [8]. The study of the Toeplitz operator is generalized to a n-dimensional structure in [9]. For the fundamental The UGC Senior research fellowship (supported by University Grants Commission, India with Ref. No. 1077/(CSIR-UGC NET DEC. 2016)) to the second author is also acknowledged. DOI: 10.21136/CMJ.2020.0088-19
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terminologies and concepts of Toeplitz and Hankel operators, one is referred to [10]. Enlightened from the work of Ho (see [5], [7]), slant Toeplitz operators are considered on L2 (Tn ) in [3]. This paper extends the study of the compression of kth-order slant Toeplitz operators to H 2 (Tn ), where the set Tn ⊂ Cn , the distinguished boundary of open unit polydisc Dn in Cn , denotes the Cartesian product of n copies of the unit circle T ⊂ C. Throughout the paper, the space of all Lebesgue measurable complex valued functions defined on Tn , which satisfies Z
Tn
|f |2 dσ < ∞,
where dσ is a normalized Lebesgue Haar measure, is denoted by L2 (Tn ). The space L∞ (Tn ) represents the space of all essentially bounded measurable functions on Tn . By the use of multiple Fourier series on Tn from the Chapter VII of [11], the space L2 (Tn ) can be expressed as L2 (Tn ) =
f : f (z1 , z2 , . . . , zn ) =
X
fm1 ,m2 ,...,mn z1m1 z2m2 . . . znmn ,
(m1 ,m2 ,...,mn )∈Zn
X
(m1 ,m2 ,...,mn )∈Zn
|fm1 ,m2 ,...,mn |2 < ∞ .
In the similar way, the space H 2 (Tn ) of n-dimensional torus Tn is given by H (T ) = f : f (z1 , z2 , . . . , zn ) = 2
n
X
fm1 ,m2 ,...,mn z1
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