On a Schur-Type Product for Matrices with Operator Entries
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On a Schur-Type Product for Matrices with Operator Entries Ismael García-Bayona1 Received: 25 September 2019 / Accepted: 22 January 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we will introduce a new Schur-type product for matrices with operator entries, and explore some of its properties. We shall see a connection between this product and the classical Schur product that will allow us to prove that this set of matrices endowed with such new product defines a Banach algebra. Also, a way to compute the operator and multiplier norms of matrices with operator entries in terms of norms of scalar matrices will be provided. As applications, we present a way to obtain multipliers for one of the products from a multiplier for the other product and show a method to construct a countable amount of elements belonging to different vector measure spaces, from a single element of L ∞ (T). Keywords Schur product · Schur multipliers · Block matrices · Toeplitz matrices · Vector-valued measures Mathematics Subject Classification 47L10 · 47A56 · 15B05 · 46G10
1 Introduction and Preliminaries The Hadamard product between two scalar matrices of the same size has been studied for over a century. The term Hadamard product was coined by von Neumann, and it was introduced in the literature by Halmos in [14]. The concept is defined as follows. Definition 1.1 (Hadamard product, ∗) Let A, B be matrices with entries in the complex or real field. Denote A = (ak, j )k, j and B = (bk, j )k, j . Then, their Hadamard product is defined as the entry-wise product:
Communicated by Ali Armandnejad. Partially supported by MTM2014-53009-P (MINECO Spain) and FPU14/01032 (MCIU Spain).
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Ismael García-Bayona [email protected] Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain
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Bulletin of the Iranian Mathematical Society
A ∗ B = (ak, j · bk, j )k, j . It is also widely known as the Schur product, since it was first investigated by Schur (see [24]). The reader can find a good exposition of the history behind the use of the names in [15]. For an historical discussion on the product, we refer to the paper of Styan [28]. Besides its theoretical interest, the importance of studying the Schur product lies on its implications and uses in many areas of mathematics, such as Banach spaces theory [3,18], operator theory [1,21], statistics [20,28], and complex function theory [23,25]. Regarding matrix analysis, we refer the reader to a recent book of Persson and Popa [22], where the Schur product becomes an important device to develop nice theories of matrix spaces parallel to their scalar counterparts. The study of matrices with entries in more general spaces (such as spaces of operators or C ∗ -algebras) as a natural generalization of matrices with scalar entries has gathered the interest of several authors (see, for example, [12,26] or the recent paper in [2]), and often, it has been in conjunction with some version of the Schur product, like in [13,19] or in more recent papers
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