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