We consider nonnegative r-potent matrices with finite dimensions and study their decomposability. We derive the precise conditions under which an r-potent matrix is decomposable. We further determine a general structure for the r-potent matrices based on their decomposibility. Finally, we establish that semigroups of r-potent matrices are also decomposable.

An 𝑟-potent matrix in 𝑀𝑛( ) is an 𝑛 × 𝑛 matrix satisfying 𝐸𝑟 = 𝐸. A multiplicative semigroup in 𝑀𝑛( ) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each 𝐴 ∈ . A semi-group of non-negative 𝑟-potent matrices in 𝑀𝑛( ) is known to be decomposable if rank (𝑆) > 𝑟 − 1 for all 𝑆 in . Further, a semigroup in 𝑀𝑛( ) of nonnegative matrices will be called a full semigroup if has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative 𝑟-potent matrices in 𝑀𝑛( ) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative 𝑟-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of 𝑟-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of 𝑟-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.

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Abstract:We investigate the decomposability of nonnegative compact r-potent operators on a separable Hilbert space L 2 pX q. We provide a constructive algorithm to prove that basis functions of range spaces of nonnegative r-potent operators can be chosen to be all nonnegative and mutually orthogonal. We use this orthogonality to establish that nonnegative compact r-potent operators with range spaces of dimension strictly greater than r´1 are decomposable.