Li–Yorke Chaos in Linear Systems with Weak Topology on Hilbert Spaces

This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator.