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Shorted operators. ii
William N. Anderson,G. E. Trapp +1 more
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In this article, the authors prove that the shorted operator exists and develop various properties, including a relation to parallel addition, including the relation between the short operator and parallel addition.Abstract:
For a positive operator A acting on a Hilbert space, the shorted operator $\mathcal{L}( A )$ is defined to be the supremum of all positive operators which are smaller than A and which have range lying in a fixed subspace S. This maximization problem arises naturally in electrical network theory. In this paper we prove that the shorted operator exists, and develop various properties, including a relation to parallel addition [Anderson and Duffin, J. Math. Anal. Appl., 11 (1969), pp. 576–594]. The basic properties of the shorted operator were developed for finite-dimensional spaces by Anderson [this Journal, 20 (1971), pp. 520–525] ; some of these theorems remain true in all Hilbert spaces, but the proofs are different.read more
Citations
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Positive solutions to X = A−BX-1 B∗
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References
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Finite-Dimensional Vector Spaces
TL;DR: The first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces, was the Finite Dimensional Vector Spaces (FDVSP) by Halmos as discussed by the authors.
Journal ArticleDOI
On majorization, factorization, and range inclusion of operators on Hilbert space
TL;DR: In this paper, it was shown that a close relationship exists between the notions of majorization, factorization, and range inclusion for operators on a Hilbert space, and that these notions fit together to yield theorems.
Journal ArticleDOI
Series and parallel addition of matrices
TL;DR: In this paper, it was shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation, and the norms are found to satisfy the inequality ∥ A : B ∥ ⩽ ∥ ∥ a ∥ : ∥ b ∥.