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STRONG SUMS OF PROJECTIONS IN VON NEUMANN FACTORS.
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"... Abstract. This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collect ..."
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Abstract. This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is “diagonalizable”. A simpler necessary and sufficient condition is given in the type III factor case. 1.
INVERTIBILITY AND FREDHOLMNESS OF LINEAR COMBINATIONS OF QUADRATIC, kPOTENT AND NILPOTENT OPERATORS
"... Abstract. Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P + Q is equivalent to that of aP + bQ for nonzero a, b with a + b 6 = 0. In this note, we obtai ..."
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Abstract. Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P + Q is equivalent to that of aP + bQ for nonzero a, b with a + b 6 = 0. In this note, we obtain a similar result for square zero operators and those operators satisfying x2 = dx for some scalar d. More generally, we show that if P,Q satisfy a quadratic polynomial (x − c)(x − d) then the linear combination aP + bQ − c(a+ b) being invertible or Fredholm (and the index) is independent of the choice of the nonzero scalars a, b. However, this is not the case when P and Q are involutions, unitaries, partial isometries, kpotents (k ≥ 3) and other nilpotents, as counterexamples are provided. 1.
2 POSITIVE COMBINATIONS OF PROJECTIONS IN VON NEUMANN ALGEBRAS AND PURELY INFINITE SIMPLE C*ALGEBRAS
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Sums of ldempotent Matrices
"... We show that any complex square matrix T is a sum of finitely many idempotent matrices if and only if tr T is an integer and tr T> rank T. Moreover, in this case the idempotents may be chosen such that each has rank one and has range contained in that of T. We also consider the problem of the min ..."
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We show that any complex square matrix T is a sum of finitely many idempotent matrices if and only if tr T is an integer and tr T> rank T. Moreover, in this case the idempotents may be chosen such that each has rank one and has range contained in that of T. We also consider the problem of the minimum number of idempotents needed to sum to T and obtain some partial results. A complex square matrix T is idempotent if T ” = T. In this paper, we characterize matrices which can be expressed as a sum of finitely many idempotent matrices and consider the minimum number of idempotents needed in such expressions. In the following, tr T denotes the trace of a matrix T, ran T denotes its range, rank T the dimension of ran T, and ker T the kernel of T. The n x n identity matrix is denoted by I,, or 1 if the size is not emphasized. Similarly for the zero matrix: 0, or 0. Two matrices T and S are similar, denoted T = S, if XT = SX for some nonsingular matrix X; they are unitarily equivalent, T z S, if the above X can be chosen to be unitary. If T and S act on spaces H and K, respectively, then acts on H @ K, the orthogonal direct sum of H and K. We start with the characterization of sums of idempotents.