ABSTRACT
We examine the Hausdorff and Quasi Hausdorff matrices which act as an operator on Hardy spaces by its action on Taylor coefficients of an analytic function. Using the integral representation of the series on the Hardy spaces, we find the conditions on the weight functions on , to determine the boundedness of the operator on weighted integrable spaces and its adjoint operator on the respective domain.
MATHEMATICS SUBJECT CLASSIFICATION:
1. Introduction
Cesàro, Euler, Taylor and Hausdorff means have been studied in connection with summability of the series and assorted function spaces like integrable spaces, Hardy spaces, Bergman spaces see for example (Alexits, Citation1953; Hardy, Citation1943; Hung et al., Citation2006; Kalaivani & Youvaraj, Citation2013; Liflyand, Citation2013), and (Kalaivani & Madhu, Citation2024).
Hardy (Hardy, Citation1920) examined Cesàro matrix act as an operator on integrable spaces and determined the norm of the operator. Hardy’s result provides wide visibility to analyzing the behavior of Cesàro operator in various spaces like the Hardy spaces on the unit disk, Bergman spaces and so on, the reader is referred to see Hardy (Citation1956), Hung et al. (Citation2000) and Siskakis (Citation1987).
Let be an infinite matrix. For an analytic function f on unit disk in the complex plane, consider the power series
and for transpose matrix of the power series
where
It is natural to address the problem of finding the conditions on the coefficients or matrix, so that for an analytic function f, the series and defined and how the matrix induces as an operator on the corresponding domain.
For the Cesàro and Hankel matrix , numerous works are done on Hardy, Bergman spaces of the unit disk.
In this paper, we consider as generalized Hausdorff matrix, since it plays a significant part in summability theory for different cases of measure ϖ and In particular, for moment sequences, we obtain Cesàro, Euler, Riesz, Taylor, Holder matrices and so on as a special case of generalized Hausdorff matrix.
Generalized Hausdorff matrix: Let and ϖ be of bounded variation on For we define
The operator obtained from and its transpose of the matrix are nothing but generalized Hausdorff operator and quasi Hausdorff operator . In particular for we obtain Hausdorff matrix as a special case of generalized Hausdorff matrix.
P. Galanopoulos and A. G. Siskakis (Galanopoulos & Siskakis, Citation2001) studied Hausdorff operator on the Hardy spaces of the unit disk and obtained the results in terms of composition operator. Later, P. Galanopoulos and M. Papadimitrakis (Galanopoulos & Papadimitrakis, Citation2006) considered Hausdorff and quasi Hausdorff matrices are act as an operator on the spaces of analytic functions on the unit disk Explicitly, they expressed the series in terms of integral as a composition function and addressed the problem on the Hardy spaces, Bergman spaces and Dirichlet space.
The action of the generalized Hardy-Cesàro operator on spaces of weighted integrable spaces have been studied by Pedersen (Pedersen, Citation2019). He briefly examined the boundedness and weakly compactness of an operator on the respective domain.
The aforementioned works motivate the study of generalised Hausdorff operator on Hardy spaces and using the integral representation of the operator in order to determine the condition on ϖ to corroborate the boundedness of the operator.
In section 2, we will examine the convergence of the power series and identify the relation between the series and composition function for the generalized Hausdorff and quasi Hausdorff operator.
In section 3 and 4, we investigate the action of the generalized Hausdorff operator and quasi Hausdorff operator on weighted spaces of integrable functions and determine the boundedness of the operator under certain conditions on weighted functions w1 and w2 on Also, we address the problem for dual of the operators and on respective domain.
2. Integral representation of the generalized Hausdorff operator
Theorem 1.
Let be an generalized Hausdorff matrix and for . Then
is analytic on
can be expressed as an integral form,
(1) (1)
Proof.
Let and Since Taylor coefficients of an analytic function are bounded by M > 0. Thus
which yields the radius of convergence for is atleast 1. Hence is an analytic function on the unit disk .
Consider
For each the series converges uniformly to Hence, by interchanging the sum and integral in the above equation,
Theorem 2.
Let be an generalized Hausdorff matrix and for . Then
is analytic on
can be expressed as an integral form,
(2) (2)
Proof.
Let and Then
which yields the radius of convergence for is atleast 1. Hence is an analytic on the unit disk .
Consider
Thus
3. Boundedness of the operator on weighted integrable spaces
In Section 2, we obtained the integral representation of and on the Hardy spaces of unit disk. Using the representation, we will study the action of generalized Hausdorff and quasi Hausdorff operator on weighted spaces of integrable functions on . Let w1 be a non-negative continuous function on and be a space of measurable functions f on with
Denote be a space of measurable functions f on with
Let be a space of locally finite, complex Borel measures ϖ on with
In the following theorem, we find the necessary and sufficient condition on weighted functions w1 and w2 for the boundedness of generalized Hausdorff operator on weighted integrable spaces on
Theorem 3.
Let w1 and w2 be the non-negative weight functions on The operator is bounded linear operator from to iff
Proof.
Let and assume the inequality holds a.e. on . Then
by the assumptions on w1 and w2 in (Equation3(3) (3) ). Note that
obtained the inequality by using Fubini’s theorem and (Equation4(4) (4) ). Thus is in and for some C > 0. Conversely, assume is bounded linear operator from to Let , then by usual inner product with , we obtain
Since which is closed and identify is the dual of , then there exists a mapping such that is measurable and bounded on for every with Thus
Equating (Equation5(5) (5) ) and (Equation6(6) (6) ), we obtain
for every As is an element of we get
where χ is the characteristic function on the respective domain. Then
By the last equality, we conclude the inequality (Equation3(3) (3) ) holds a.e. on if the operator is bounded on weighted spaces of the integrable functions from to
Theorem 4.
Let be a bounded linear operator from to such that the condition (Equation3(3) (3) ) holds a.e. on Then there exists an adjoint operator such that a.e. on
Proof.
For and we have
where a.e. on Thus is an adjoint operator of on Also, for we have
the last inequality obtained by the boundedness of which concludes the proof the theorem.
4. Boundedness of the operator on weighted integrable spaces
Theorem 5.
Let w1 and w2 be the non-negative weight functions on The operator is bounded linear operator from to iff
Proof.
by the assumption (Equation7(7) (7) ), we obtained is bounded operator from to Conversely, assume is bounded linear operator on the respective domain. Let then
where
Similar to the steps as in Theorem 3, we obtain
for every Equating (Equation8(8) (8) ) and (Equation9(9) (9) ), we obtain
As we have
from the notation of which concludes the proof of the theorem.
Theorem 6.
Let be a bounded linear operator from to such that the condition holds a.e. on Then there exists an adjoint operator such that
Proof.
For and we have
where
Thus is an adjoint operator of on Also, for and we have
For we have
the last two inequalities (Equation11(11) (11) ) and (Equation12(12) (12) ) obtained by the boundedness of
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References
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