Decomposability and structure of nonnegative bands in ℳn(ℝ)

DECOMPOSABILITY OF NONNEGATIVE r-POTENT OPERATORS ON L^2(X)

“…An operator A on the Hilbert space L 2 pX q is said to be decomposable [1,2] if there exists a nontrivial standard subspace of L 2 pX q that is invariant under A. Marwaha [3] showed that nonnegative idempotent operators of rank greater than one are decomposable in finite dimensions. Marwaha [4] further established that nonnegative idempotent operators with range spaces of dimension greater than one are decomposable in infinite dimensions.…”

“…An operator A on the Hilbert space L 2 pX q is said to be decomposable [1,2] if there exists a nontrivial standard subspace of L 2 pX q that is invariant under A. Marwaha [3] showed that nonnegative idempotent operators of rank greater than one are decomposable in finite dimensions. Marwaha [4] further established that nonnegative idempotent operators with range spaces of dimension greater than one are decomposable in infinite dimensions. The results in [3] (finite-dimension case) for idempotent operators were generalised by Thukral and Marwaha to r-potent operators in [5] (recall that an operator A is said to be r-potent [6] if A r " A, where r is a positive integer).…”

“…Marwaha [4] further established that nonnegative idempotent operators with range spaces of dimension greater than one are decomposable in infinite dimensions. The results in [3] (finite-dimension case) for idempotent operators were generalised by Thukral and Marwaha to r-potent operators in [5] (recall that an operator A is said to be r-potent [6] if A r " A, where r is a positive integer). Specifically, Thukral and Marwaha [5] showed that all nonnegative r-potent operators of rank greater than r´1 are decomposable in finite dimensions.…”

“…Specifically, Thukral and Marwaha [5] showed that all nonnegative r-potent operators of rank greater than r´1 are decomposable in finite dimensions. A similar generalisation from idempotent operators to r-potent operators in the infinite-dimension case [4] however does not exist in the literature. In this paper, our main contribution is to provide this generalisation of results in [4] for idempotent operators to the r-potent case.…”

“…A similar generalisation from idempotent operators to r-potent operators in the infinite-dimension case [4] however does not exist in the literature. In this paper, our main contribution is to provide this generalisation of results in [4] for idempotent operators to the r-potent case. It should be noted that proof of decomposability for r-potent operators in the finitedimension case [5] depends critically on the Perron-Frobenius Theorem [7] for matrices.…”