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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