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Transactions of the American Mathematical Society

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Schur products of operators and the essential numerical range


Author: Quentin F. Stout
Journal: Trans. Amer. Math. Soc. 264 (1981), 39-47
MSC: Primary 47B37; Secondary 47A05, 47A10, 47D99
DOI: https://doi.org/10.1090/S0002-9947-1981-0597865-2
MathSciNet review: 597865
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Abstract: Let $ \mathcal{E} = \{ {e_n}\} _{n = 1}^\infty $ be an orthonormal basis for a Hilbert space $ \mathcal{H}$. For operators $ A$ and $ B$ having matrices $ ({a_{ij}})_{i,\;j = 1}^\infty $ and $ ({b_{ij}})_{i,\;j}^\infty = 1$, their Schur product is defined to be $ ({a_{ij}}{b_{ij}})_{i,\:j}^\infty = 1$. This gives $ \mathcal{B}(\mathcal{H})$ a new Banach algebra structure, denoted $ {\mathcal{P}_\mathcal{E}}$. For any operator $ T$ it is shown that $ T$ is in the kernel (hull(compact operators)) in some $ {\mathcal{B}_\mathcal{E}}$ iff 0 is in the essential numerical range of $ T$. These conditions are also equivalent to the property that there is a basis such that Schur multiplication by $ T$ is a compact operator mapping Schatten classes into smaller Schatten classes. Thus we provide new results linking $ \mathcal{B}(\mathcal{H})$, $ {\mathcal{B}_\mathcal{E}}$ and $ \mathcal{B}(\mathcal{B}(\mathcal{H}))$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0597865-2
Keywords: Schur multiplication, Hadamard multiplication, essential numerical range, matrix representations
Article copyright: © Copyright 1981 American Mathematical Society

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