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Choo - Whan Kim

T^g' = T^g e L°° (X, ju), where g'=l yg- Thus we have as м -► oo,

\\Qnng - Pg\\i = \\QnT^g'~Pg'\\i = \\QnT^g'-PT^g'\\i--o,

This proves the second part of the theorem.

By a sub-Markov operator P with kernel p(x, y) we mean P e Jt' of the form

^^ W = \x Pix, y)g{y)d^{y) {g e L-),

where p{x, y) is a nonnegative measurable function on the product space X^ = X xX such that jx P(^? y)d/x(y)^ 1. Such an operator is also called a /г-contmuous sub-Markov operator. Furthermore, we say that the operator P is of Hilbert- Schmidt type if its kernel p{x,y) belongs to L^(X^,/i^), ß^=fiXfi. Corollary of Lemma 2, together with minor modifications in the proofs of Theorems 3 and 4 of [5, pp. 212, 213] proves the following results.

Theorem 7. For each ß-continuous sub-Markov operator P there exists a sequence(К^) inch(!F') such that \\P— i^fcH oo,i -^0 ^^ /c-^oo.

Theorem 8. For each sub-Markov operator P of Hilbert-Schmidt type, there exists a sequence (K^) in ch(^') such that || F— K^ || 2 ->0 as /c-> 00.

Similarly , utilizing Lemma 4 we show easily the following results.

Theorem 9. For each fi-continuous doubly substochastic operator F, there exists a sequence (Rj,) in сЬ(Ф') such that || F—К^|| ^ д -^0 as /c-> 00.

Theorem 10. For each doubly substochastic operator P of Hilbert-Schmidt type, there exists a sequence (F^) in сЬ(Ф') such that || F— F;,|| 2-^0 as /c-> oc.

References

1 Brown, J R Approximation theorems for Markov operators Pacific J Math 16,13—23(1966)

2 Dunford, N, Schwartz, J T Linear operators, Part I New York Interscience 1967

3 Kadison,R V The trace in finite operator algebras Proc Amer Math Soc 12,973—977(1961)

4 Kim,C W Uniform approximation of doubly stochastic operators Pacific J Math 26,515—527 (1968)

5 Kim, С W Approximation theorems for Markov operators Z Wahrscheinhchkeitstheorie verw Geb 21,207—214(1972)

6 Kim,C W Approximations of positive contractions on L* [0,1] Z Wahrscheinlichkeitstheorie verw Geb 24, 335—337 (1972)

7 Kim, С W Approximations of positive contractions onf^(p=i, 00) (preprint)

8 Mirsky, L Results and problems m the theory of doubly-stochastic matrices Z keitstheorie verw Geb 1,319—334(1963)

9 Revesz,P Aprobabilisticsolutionof problem 111 of G Birkhoff Acta Math Acad Sci Hungar 13, 187—198(1962)

10 Phelps, R R . Extreme positive operators and homomorphisms Trans Amer Math Soc 108, 265—274(1963)

11 Royden, H L Real analysis, 2nd ed New York McMillan 1968

Choo - Whan Kim

Department of Mathematics

Simon Fraser University

Burnaby , British Columbia V5A 1S6, Canada

( Received February 3, 1975)