Asymptotic properties for positive and Toeplitz matrices and operators. (English) Zbl 0701.47012

Adamyan, V. M.

Linear operators in function spaces, 12th Int. Conf. Oper. Theory, Timişoara/Rom. 1988, Oper. Theory, Adv. Appl. 43, 17-38 (1990).

[For the entire collection see Zbl 0687.00009.]
Let \(J=(J_{ik})^{\infty}_ 1\) be an infinite positive matrix, i.e. for all N the blocks \(J_ N=(J_{jk})^ N_ 1\) are positive definite. Then the sequence of the numbers \(d_ N=\sum^{N}_{i,k=1}(J_ N^{- 1})_{ik}\) is not decreasing. The author studies the asymptotic behaviour of this sequence. For an infinite positive Toeplitz matrix J he computes under certain conditions the limit of \(\{d_ N\}\), \(\{(1/N)d_ N\}\), \(\{\) (1/N) trace \(J_ N^{-1}\}\) and \(\{\) (1/N) \(\ell n(\det J_ N)\}\). Similar results are obtained for multidimensional Toeplitz matrices generated by positive definite functions.

Reviewer: K.-H.Förster

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A15	Determinants, permanents, traces, other special matrix functions
15A90	Applications of matrix theory to physics (MSC2000)
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics

Keywords:

infinite positive matrix; blocks; asymptotic behaviour; infinite positive Toeplitz matrix; multidimensional Toeplitz matrices generated by positive definite functions