On the spectrum of positive linear operators with a partition of unity property

Abstract

We characterize the spectrum of positive linear operators between Banach function spaces having finite rank and a partition of unity property. Our main result states that all the points in the spectrum are eigenvalues and 1 is the only eigenvalue on the unit circle. Finally, we show that the iterates converge in the uniform operator topology to a projection operator that reproduces constant functions and we provide a simple criterion to obtain the limiting projection operator.We study positive linear operators that have finite rank on a general infinite-dimensional complex Banach function space X that contains the constant function 1 with norm equal to one. In addition, we assume that the associated basis functions of the positive finite-rank operator form a partition of unity. Operators of this kind are used in many applications to approximate functions where only a finite number of samples are available. The partition of unity property guarantees the exact reconstruction of constant functions. Of our interest here is the asymptotic behaviour of iterative applications of the operator and the question whether the limit exists.The asymptotic behaviour of the iterates of positive linear operators has extensively been discussed by many authors. Kelisky and Rivlin [12] have first been considering the limit of iterates of the classical Bernstein operator on the space C ([0, 1]). This result has been extended by Karlin and Ziegler [10] to a more general setting. In [15,16], J. Nagel has examined the asymptotic behaviour of the Bernstein and the Kantorovič operators. Using a contraction principle, Rus [19] has shown an alternative way to prove the convergence of the iterates of the Bernstein operator. The iterates of the Bernstein operators have been also revisited by Badea [2] using spectral properties. Recently, contributions have been made by and by Altomare [1] using methods based on Korovkin-type approximation theory. However, all these results are restricted to the space of continuous functions, i.e., are not applicable for the L p spaces, and there is no general theory that guarantees the existence of the limit of the iterates.

Lower estimates for linear operators with smooth range

“…n n , while this property also follows by [18]. The second largest eigenvalue γ n of B n is γ n := λ 2,n = n−1 n .…”

“…We have that ker(V m ∆n,k − I) = span {1} and D1 = 0 holds. By [18], we can conclude that σ(V ∆n,k ) ⊂ B(0, 1) ∪ {1} , holds. The operator norm of the differential operator D, can be obtained similarly to the Kantorovič operator.…”

Iterates of Markov operators and their limits

“…In the following, we want to answer the question whether the limit of the iterates T m for m → ∞ exists and if so to which operator the iterates converge. In Nagler [18] is has been shown that the partition of unity property, which is here equivalent to the ability to reproduce constant functions, guarantees that σ(T ) ⊂ B(0, 1) ∪ {1}. To apply our main result, we have to specify the fixed point spaces of T and its adjoint T * .…”

An answer to a conjecture on the limit of the iterates of Jackson type operator Gs, n

“…Their approach uses the result of C. Badea [1]. In [9] Nagler provide a simple criterium to obtain the limiting projection operator. Our study in [18] was inspired by Badea [1], where the asymptotic behavior of the iterates is characterized by spectral properties of the operator.…”