NEW ZEALAND JOURNAL OF MATHEMATICS Volume 34 (2005), 43 60 ON MARKOV AND KOLMOGOROV MATRICES AND THEIR RELATIONSHIP WITH ANALYTIC OPERATORS N. Katilova (Received February 2004) Abstract. In this article, the relationship between sub Markov and sub Kolmogorov matrices is investigated. In addition some results about analytic operators and analytic semigroups are exhibited. In particular it is shown that analyticity properties have ramifications for limiting properties of bounded analytic semigroups. These result has some interesting consequences for matrices with Markov and Kolmogorov properties.. Introduction First we introduce some notions and definitions which we are going to use. Definition.. The matrix K = (k ij ) n i,j= is called a Kolmogorov matrix, if the following two conditions are fulfilled: and k ij 0, i j and i, j =,..., n () n k ij = 0, j =,..., n. (2) The matrix K is called a sub-kolmogorov matrix if it satisfies () and if (2) is replaced with n k ij 0, j =,..., n. (3) For matrices we introduce the notion of the (positive) maximum principle, as well as the sub-markov property. In Van Casteren [6] the reader may find some related results for linear operators which satisfy the maximum principle. Definition.2. The matrix K = (k ij ) n i,j= is said to satisfy the positive maximum principle if it possesses the following property: for every row vector f = (f,..., f n ) with max i n f i > 0 the inequality (fk) i0 0 (4) is satisfied whenever the index i 0 is chosen in such a way that f i0 = max i n f i. 99 Mathematics Subject Classification 60G46, 60J25. Key words and phrases: Markov processes, ergodicity conditions. The author is grateful to the University of Antwerp (UA) for its material support; she also thanks Jan A. Van Casteren for his advise and stimulating discussions.
44 N. KATILOVA The matrix K = (k ij ) n i,j= is said to satisfy the maximum principle if it possesses the following property: for every row vector f = (f,..., f n ) the inequality (fk) i0 0 (5) is satisfied whenever the index i 0 is chosen in such a way that f i0 = max i n f i. Definition.3. A matrix M = (m ij ) n i,j= is called a sub Markov matrix, if the following two conditions are fulfilled: and m ij 0, i, j =,..., n (6) n m ij, j =,..., n. (7) Definition.4. Let K be a square matrix. (a) An eigenvalue µ of K is called dominant if lim t e tk (I P ) = 0. Here P is the Dunford projection on the generalized eigenspace corresponding to µ; i.e. P = 2πi γ (λi K) dλ, where γ is a (small) positively oriented circle around µ. The disc centered at µ and with circumference γ does not contain other eigenvalues. (b) An eigenvalue µ of K is called critical if it is dominant and the zero space of K µi is one dimensional. Remark.5. Here the generalized eigenspace is the set of all vectors x C n for which (µi K) l x = 0 for some l n. Notice that n k=0 α k (µi K) l = 0 for appropriately chosen constants α k, 0 k n. In the sequel we consider the following simplex W in space R n : { } n W = x R n : x 0,..., x n 0, x i =, (8) and the following (n ) dimensional subspace L of R n : { } n L = x R n : x i = 0. (9) 2. The Relationship Between Markov and Kolmogorov Matrices We consider a time continuous system with a constant matrix K: ẋ(t) = Kx(t), < t <. (0) Let K be a Kolmogorov matrix. Then 0 is a dominant eigenvalue of K; let P : R n R n be the Dunford projection on the eigen space corresponding to the eigenvalue 0. If the eigenvalue 0 has multiplicity, then P projects on the one dimensional subspace R. Since 0 is a dominant eigenvalue of K, a key spectral estimate of the following form is valid: fe tk (I P ) Ce δt f, f R n, () where δ is strictly positive, is some (chosen) norm on R n, and where C is some finite constant which depends on the chosen norm.
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 45 In the next proposition we want to exhibit the relationship between matrices which satisfy the maximum principle, which possess the sub Markov property, and which preserve positivity. Proposition 2.. Let K be an n n matrix. (a) Suppose that K satisfies the maximum principle. Then the following inequalities hold true: max (f i s (fk) i n i ) max f i; (2) i n min (f i s (fk) i n i ) min f i, (3) i n where the row vector (f,..., f n ) is arbitrary. Consequently, if s > 0 the null space of the operator f f (I sk) is trivial, and since we are in the finite-dimensional situation I sk is invertible. Moreover, if s > 0 then (I sk) ; (4) (I sk) 0. (5) (b) If K satisfies the positive maximum principle, then for s > 0 the inverses (I sk) exist. Moreover, (4) and (5) are satisfied as well. (c) The following assertions are equivalent: (i) For every vector f with 0 f the inequalities 0 f (I sk) are valid for all s > 0 for which the inverse (I sk) exists. (ii) For every vector f with 0 f the inequalities 0 fe sk are valid for all s 0. (iii) K satisfies the positive maximum principle. (iv) K satisfies both the inequalities (4) and (5). Remark 2.2. As indicated in (c) the inequalities (4) and (5) can be combined. If 0 s K <, then I sk is invertible. The following equalities represent the matrix e sk, s > 0, as certain limits in terms of (I tk), where t > 0 is small: ( fe sk = lim f I s ) n n n K = lim f n e ns e ns(i n K) = lim f n esk(i n K) = lim n e ns (ns) k f k! k=0 The implication (ii) (i) follows from the useful representation: f (I sk) = 0 e t ( f e stk) dt. ( I n K ) k. Proof. (a) Choose i 0 n in such a way that f i0 = max {f i : i n}. Then (fk) i0 0, and thus ( ) fi s(fk) i fi0 s(fk) i0 f i0 = max {f i : i n}. max i n
46 N. KATILOVA This proves inequality (2). Upon replacing f with f inequality (3) follows from (2).( Upon writing g = f(i sk), and thus f = g(i sk) (2) implies max ) i n g(i sk) max i i n g i. With g =, (4) follows. In the same manner (5) is a consequence of (3). (b) Let k ij be the entries of the matrix K. Our first step is to show that the positive maximum principle implies that K is sub Kolmogorov; i.e. k ii0 0 for i i 0, and n k ii 0 0, for i 0 n. But this is clear since 0 < = max i n i = i0, and hence n k ii 0 = ( K) i0 0. Next we fix ε > 0 arbitrary, and we suppose that i i 0, i, i 0 n. Define the vector f ε as follows: f ε,j =, for j = i, f ε,j = ε, for j = i 0, and f ε,j = 0, elsewhere. Then f ε,j is minimal and strictly negative at j = i 0. From the positive maximum principle it follows that k ii0 εk i0i 0 = (f ε K) i0 0. Hence k ii0 εk i0i 0 for every ε > 0. Consequently, k ii0 0, for i i 0. Suppose f (I sk) 0. Assume f i0 = min i n f i < 0. Then, by the positive maximum principle (fk) i0 0, and hence n 0 (f sfk) i0 = f i0 s f i k ii0 = f i0 + s n n (f i0 f i ) k ii0 sf i0 (the matrix K is sub Kolmogorov and f i0 < 0) k ii0 f i0 < 0, (6) which is a contradiction. Hence, f (I sk) 0 implies f 0. In particular, if f (I sk) = 0, then f = 0. As a result, we see that the null space of I sk is zero, whenever s > 0, and so the operator f f (I sk) is invertible. Consequently, (I sk) 0. Next let the row vector f be such that f (I sk) =. Then, by what we proved above f 0. Moreover, assume to arrive at a contradiction that, for some i 0 n, f i0 = max i n (f i ) > 0. Then 0 s ( (f )K ) i 0 = s (fk) i0 s ( K) i0 = f i0 s ( K) i0 f i0 > 0, (7) which is a contradiction. Hence, the equality f (I sk) = implies 0 f. Consequently, 0 (I sk). All this proves assertion (b). (c) (i) (ii). For the implication (i) (ii) we write ( fe sk = lim f I s ) n n n K. The implication (ii) (i) follows from the representation: f (I sk) = 0 e t ( f e stk) dt. The implication (iii) (iv) is proved in (b). The implication (iv) (i) is trivial.
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 47 (ii) (iii). Let the vector f be such that max j n f j = f j0 > 0. We have to prove (fk) j0 0. First we notice that f (max j n f j ), and hence by (ii) fe sk (max j n f j ) e sk, s > 0. Consequently we have (fk) j0 = ( ) ( fe sk f j j0 max j n f j e sk f lim 0 j j0 lim 0 s 0 s s 0 s ( ( f ) j0 e sk j0 f j0 ( ) = lim lim = 0, (8) s 0 s s 0 s where we used f j0 0 and the inequality e sk, which is a consequence of (ii). This proves the implication (ii) (iii). All this proves assertion (c), and thus Proposition 2. follows. The following theorem shows that matrix operators which satisfy the positive maximum principle are intimately related to sub Markov chains. Theorem 2.3. Let K be an n n matrix. The following assertions are equivalent: (i) the matrix K satisfies the positive maximum principle; (ii) for every t > 0 the matrix e tk is sub Markovian; (iii) K is a sub Kolmogorov matrix. Proof. Let t > 0 be arbitrary. Representing e tk as e tk = lim (I tn ) n K n (4) implies e tk, and (5) implies e tk 0. Altogether this shows the implication (i) (ii). Conversely, let f = (f,..., f n ) be an arbitrary row vector in R n. Choose i 0 n in such a way that m := max i n f i = f i0. Then f m, and so ( (fe tk (fk) i0 = lim ) ) t 0 i0 t f ( i 0 lim m ( e tk) ) t 0 t i 0 m (the matrix e tk is sub-markov) m ( ) m lim ()i0 () t 0 t i0 lim 0 = 0. (9) t 0 t This shows the implication (ii) (i). Next we prove (ii) (iii). By (ii) we know that, for all t > 0, e tk. Since ( K = lim e tk ) 0, t 0 t it follows that K 0. This proves part of the sub Kolmogorov property. We still have to show that k ij 0, for j i. This assertion follows from the equality (δ ij is the Kronecker delta symbol): ( k ij = lim (e tk ) t 0 t ij δ ) ij, which for j i shows that k ij 0, since, by assumption, the matrices e tk, t > 0, have the sub-markov property, and thus have non-negative entries.
48 N. KATILOVA (iii) (ii). Again we use the following representation for e tk : ( e tk = lim I t ) n n n K. (20) From the sub Kolmogorov property it follows that (I sk), s > 0, is positivity preserving. For, let the row vector f = (f,..., f n ) be such that f (I sk) 0, and assume that f i0 = min i n f i < 0. Since K 0, we have for s > 0 n 0 f i0 s (fk) i0 = f i0 s f i k ii0 = f i0 + s n n (f i0 f i ) k ii0 sf i0 k ii0 f i0 sf i0 ( K) i0 f i0 < 0, (2) which is a contradiction. Consequently, (I sk) 0. Moreover, since K 0, we also obtain: (I sk) = s ( K) (I sk) 0. (22) Finally, we obtain from (22) together with (20) and the fact that, for s > 0 small enough, the operator (I sk) is positivity preserving, the operators e sk, s 0, have the sub Markov property. The following proposition establishes the relationship between matrices with the Kolmogorov property and matrices with the Markov property. Proposition 2.4. Let K = (k ij ) n i,j= be an n n matrix. The following assertions are equivalent: (i) The matrix K has the Kolmogorov property; (ii) For every t > 0 the matrix e tk possesses the Markov property. Proof. For completeness we insert a proof. (ii) (i). By hypothesis we know that, for all t > 0, e tk =. Since ( K = lim e tk ) = 0, t 0 t it follows that K = 0. This proves part of the Kolmogorov property. We still have to show that k ij 0, for j i. This follows from Theorem 2.3 implication (ii) (iii). (i) (ii). Here we use the following representation for e tk : ( e tk = lim I t ) n n n K. (23) Since K has the Kolmogorov property, we see (I sk) =, and hence ( = lim I t ) n n n K = e tk. (24) The positivity of the entries of the matrix e tk again follows from Theorem 2.3 implication (iii) (ii).
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 49 3. Analytic Operators and Semigroups The following (important) Theorems 3. and 3.6 are inspired by ideas in Nagy [5]. Theorem 3.. Let M be a bounded linear operator in a Banach space X. By definition the sub-space X 0 of X is the closure of the vector sum of the range and zero space of I M: X 0 = R (I M) + N (I M). Suppose that the spectrum of M is contained in the open unit disc union {}. The following assertions are equivalent: ( (i) sup λ < λ) (I λm) x < for every x X 0 ; (ii) sup n N M n x < and sup n N (n + ) M n (I M) x < for every x X 0 ; (iii) sup t>0 e t(m I) x < and supt>0 t (M I) e t(m I) x < for every x X 0 ; (iv) There exists 2π < α < π such that { } sup λ (λi (M I)) x : α < arg(λ) < α <, for all x X 0 ; (v) There exists 2π < α < π such that { } (I sup M) ((λ + ) I M) x : α < arg(λ) < α <, for all x X 0 ; (vi) For every x X 0 the following limits exist P x := lim n M n x and (I P ) x = lim re iϑ 0<r< (vii) For every x X 0 the following limit exists (I P ) x := lim re iϑ 0<r< (I M) ( I re iϑ M ) x. (25) (I M) ( I re iϑ M ) x. (26) Moreover, if M satisfies one of the conditions (i) through (vii), then X 0 = R (I M) + N (I M). Remark 3.2. The Banach Steinhaus theorem implies that in (i) through (v) in Theorem 3. the vector norms may be replaced with the operator norm restricted to X 0 ; i.e. the operator M must be restricted to X 0. Conditions (a) and (b) of the following corollary are satisfied if the space X is reflexive. Corollary 3.3. Let M be a bounded linear operator in a Banach space (X, ). As in Theorem 3. let X 0 be the closure in X of the sub space R (I M)+N (I M). Suppose that, for 0 < λ <, the inverse operators (I λm) exist and are bounded, and that sup 0<λ< ( λ) (I λm) <. If one of the following conditions:
50 N. KATILOVA (a) the zero space of the operator (I M), which is a sub space of the bidual space X is in fact a subspace of X; (b) the σ (X, X) closure of R ((I M) ) coincides with its closure; (c) the range of I M is closed in X; is satisfied, then the space X 0 coincides with X, and hence all assertions in Theorem 3. are equivalent with X replacing X 0. Remark 3.4. If sup n N M n <, then sup 0<λ< ( λ) (I λm) <. As can be seen from the proof the assertions (i) through (v) in Theorem 3. are also equivalent if X replaces X 0. Definition 3.5. An operator M which satisfies the equivalent conditions (i) through (v) with X replacing X 0 of Theorem 3. is called an analytic operator in the subspace X. If it satisfies the equivalent conditions (i) through (vii) of Theorem 3., then it is called an analytic operator in the subspace X 0. Proof of Corollary 3.3. If the range of I M is closed, then by the closed range theorem, the range of I M is weak -closed and hence (c) implies (b). We will prove that (a) as well (b) implies X 0 = X. First we assume (a) to be satisfied. Pick x X, and consider x = (I M) (I λm) x + ( λ) M (I λm) x = x x λ + x λ, (27) where x λ = ( λ) M (I λm) x. Then sup 0<λ< x λ <, and consequently the family x λ, 0 < λ <, has a point of adherence x in X ; i.e. x belongs to the σ (X, X ) closure of the subset {x λ : η < λ < }, and this for every 0 < η <. Fix x X. Then ( λ)m(i λm) x, (I M) x = ( λ)(i M)(I λm) x, M x ( λ) (I M)(I λm) x M x. (28) Since sup 0<λ< ( λ) (I λm) <, the identity (I M) (I λm) = λ ( I ( λ)(i λm) ) yields that sup 0<λ< (I M) (I λm) <. Consequently (28) implies x, (I M) x = lim xλ, (I M) x λ = lim λ ( λ) (I M) (I λm) x, M x = 0. (29) Hence x annihilates R ( (I M) ) and so it belongs to the zero space of the operator (I M). By assumption this zero space is a subspace of X. We infer that the vector x can be written as x = x x + x, where x is a member of N (I M), and where x x belongs to the weak closure of the range of I M. However this weak closure is the same as the norm closure of R (I M). Altogether this shows X = X 0 = closure of R (I M) + N (I M).
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 5 Next we assume that (b) is satisfied. Let x 0 be an element of X which annihilates X 0 ; i.e. which has the property that x, x 0 = 0 for all x X 0. Then x 0 annihilates R (I M), and hence it belongs to zero space of (I M). Since x 0 also annihilates the zero space of I M, it belongs to the weak closure of R ((I M) ). By assumption (b), we see that x 0 is a member of its norm closure; i.e. x 0 belongs to the intersection N ( (I M) ) R ((I M) ). We will show that x 0 = 0. By the Hahn Banach it then follows that X 0 = X. Since x 0 belongs to the closure of R ((I M) ), it follows that x 0 = lim λ (I M) ((I λm) ) x 0. (30) To see this we first suppose that x 0 = (I M) x. Then (I M) x (I M) ((I λm) ) (I M) x = ( λ) M ((I λm) ) (I M) x. (3) Since the family M ((I λm) ) (I M) x, 0 < λ <, is bounded, we see that (30) is a consequence of (3) provided x 0 belongs to the range of (I M). By the uniform boundedness of the family (I M) ((I λm) ), 0 < λ <, the same conclusion is true if x 0 belongs to the closure of the range of (I M). Since, in addition, x 0 is a member of N ((I M) ), it follows that x 0 = 0. This proves Corollary 3.3. Proof of Theorem 3.. (i) = (ii). Fix 0 < r <. The following representations from Lyubich [4] are being used: (n + )M n = ( λ) 2 (I λm) 2 dλ 2πi λ n+ ( λ) 2, (32) Put C := sup The choice r 2 = λ =r 2 (n + )(n + 2)M n (I M) = ( λ) 2 (I M) (I λm) 3 dλ 2πi λ n+ ( λ) 2 λ =r = ( λ) 2 (I λm) 2 2πi λ =r λ n+2 ( λ) 2 dλ 2πi λ =r { ( λ) (I λm) X0 : λ < (n + ) M n C 2 X0 r n 2π ( λ) 3 (I λm) 3 λ n+2 2 dλ. (33) ( λ) }. From (32) we infer π π C2 re iϑ 2 dϑ = r n r 2. (34) n n+2 yields M n X0 2 3 ec2. (35)
52 N. KATILOVA In the same spirit from (33) we obtain M 2 (n + )(n + 2) n (M I) ( X0 C 2 + C 3) r n+ r 2. (36) The choice r 2 = n+ n+3 yields the inequality: (n + ) M n (M I) 4e ( X0 C 2 + C 3). (37) 3 This proves the implication (i) = (ii). (ii) = (iii). The representations (see Nagy [5]) e t(m I) = e t k=0 t k k! M k and t (M I) e t(m I) = e t show that (iii) is a consequence of (ii). k=0 t k+ M k (M I) k! (iii) = (iv). This is a (standard) result in analytic operator semigroup theory: see e.g. Van Casteren [6], Chapter 5, Theorem 5.. (iv) = (v). The equality (I M) ((λ + ) I M) = I λ (λi (M I)) shows the equivalence of (iv) and (v). (v) = (i). Fix x X 0. The choice λ = + e iϑ = 2i sin ( 2 ϑ) e 2 iϑ, ϑ 2α, yields the boundedness of the function ϑ (I M) ( I e iϑ M ) x on the interval [ α, α]. Since, for λ =, λ, the function λ (I M) (I λm) x is continuous, it follows that the function λ (I M) (I λm) x is bounded on the unit circle. The maximum modulus theorem shows that this function is bounded on the unit disc, which is assertion (i). (i) = (vi). Fix x X 0. For 0 < r < and ϑ R we also have ( ( I P re iϑ )) (I M) x = π r 2 2π π 2r cos (ϑ t) + r 2 (I M) ( I e it M ) (I M) x dt = (I M) ( I re iϑ M ) (I M) x. (38) In (38) we use the continuity of the boundary function to show that lim re iϑ, 0 r< e it (I M) ( I e it M ) (I M) x (39) ( I P (re iϑ ) ) (I M) x = (I P ) (I M) x = (I M) x (40)
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 53 exists, and that I P is bounded projection. From (i) it follows that the function λ (I M) (I λm) x is uniformly bounded on the unit disc, and hence that the limit in (40) exists for all y in the closure of R(I M). In addition, for such vectors y we have (I P ) y = y. The limit in (40) trivially exists for x X such that Mx = x, we conclude that the limit in (i) exists for all x X 0, because x = (I P )x + P x, where (I P )x belongs to the closure of the range of I M and where P x = x (I P )x = x lim λ (I M) (I λm) x = lim λ ( λ) M (I λm) x. (4) From (4) it follows that (I M) P x = 0. In addition, from (ii), which is equivalent to (i), we see that lim n M n y = 0 for all y in the range of I M; here we use the boundedness of the sequence (n + ) M n (I M), n N. The boundedness of the sequence M n, n N, then yields lim n M n y = 0 for y R(I P ), because the range of I M is dense in the range of I P. An arbitrary x X 0 can be written as x = (I P ) x + P x. From the previous arguments it follows that lim n M n x = P x. Fix x X 0. Altogether this shows the implication (v) = (vi), provided we show the continuity of the function in (39) in the sense that lim t 0 (I M) ( I e it M ) (I M) x = (I M) x. However, this follows from the identity (I M) ( I e it M ) (I M) x (I M) x = ( e it ) (I M) ( I e it M ) Mx, together with the unform boundedness (in 0 < t π) of the family of operators: (I M) ( I e it M ). In the latter we use the (proof of the) implication (v) = (i). The implication (vi) = (vii) being trivial there remains to be shown that (vii) implies (i). For this purpose we fix x X 0 and we consider the continuous function on the closed unit disc, defined by (I M) (I λm) x for λ, λ, F (λ)x := (I P )x = lim λ (I M) (I λm) x for λ =. λ < (42) From (vii) it follows that the function F (λ)x is well defined and continuous. Hence it is bounded. The theorem of Banach Steinhaus then implies (i). Theorem 3. has the following analogue in the continuous time setting. Theorem 3.6. Let K be a closed linear operator with a dense domain in a Banach space X. Let X 0 be the closed sub space of X defined as the closure of the vector sum of the range and zero space of K: X 0 = R (K) + N (K). Suppose that the spectrum of K is contained in the open left half plane union {0}. The following assertions are equivalent:
54 N. KATILOVA λ (i) sup Reλ>0 (λi K) x < for all x X 0 ; (ii) sup t>0 e tk x < and supt>0 tke tk x < for all x X0 ; (iii) There exists 2 π < α < π such that sup α arg(λ) α λ (λi K) x < for all x X 0 ; (iv) For every x X 0 the following limits exist: and P x := lim e tk x, (I P ) x = lim λ (λi K) x, t λ 0 Reλ>0 x = (v) For every x X 0 the following limits exist: and lim λ (λi K) x; λ 0 Reλ>0 (I P ) x = lim λ 0, Reλ>0 λ (λi K) x, x = lim λ (λi λ, Reλ>0 K) x. Moreover, if K satisfies one of the conditions (i) through (v), then X 0 = R (K) + N (K). Remark 3.7. The Banach Steinhaus theorem implies that in (i) through (iii) in Theorem 3.6 the vector norms may be replaced with the operator norm restricted to X 0 ; i.e. K must be restricted to X 0 D (K). Conditions (a) and (b) of the following corollary are fulfilled if the space X is reflexive. Corollary 3.8. Let K be a closed densely defined linear operator in a Banach space (X, ). As in Theorem 3.6 let X 0 be the closure in X of the sub space R (K) + N (K). Suppose that, for 0 < λ, the inverse operators (λi K) exist and are bounded, and that sup 0<λ λ (λi K) <. If one of the following conditions: (a) the zero space of the operator (K), which is a sub space of the bidual space X, is in fact a subspace of X; (b) the σ (X, X) closure of R (K ) coincides with its closure; (c) the range of K is closed in X; is satisfied, then the space X 0 coincides with X, and hence all assertions in Theorem 3.6 are equivalent with X in place of X 0. Remark 3.9. If sup t>0 e tk <, then sup0<λ λ (λi K) <.
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 55 The proof of Corollary 3.3 can be adapted to yield the result in Corollary 3.8. Instead of (27) we exploit an identity of the form: x = K (λi K) x + λ (λi K) x. Definition 3.0. A densely defined closed linear operator K satisfying one (and hence all) of the conditions (i) through (iii) in Theorem 3.6 with X instead of X 0 is called the generator of a bounded analytic semigroup in X 0 : see Blunck []. Proof of Theorem 3.6. The proof of Theorem 3.6 follows the same pattern as that of Theorem 3.. Among other things we use the following equalities: te tk = 2πi ω+i ω i λ 2 e tλ (λi K) 2 λ 2 dλ; 2 t2 Ke tk = 2πi ω+i ω i λ 2 e tλ K (λi K) 3 λ 2 dλ; (I P (ω + iξ)) K = ω π ω 2 + (t ξ) 2 ( K) (iti K) K dt = K ((ω + iξ) I K) K. (43) First one shows the equivalence of the assertions in (i) through (iii) in Theorem 3.6: this is a standard result in analytic semigroup theory. In fact, the equivalence of (i), (ii) and (iii) is also true if X replaces X 0. (i) = (iv). In order to prove the implication (i) = (iv) in (43) we let ω + iξ tend to 0, and we use the continuity of the boundary function t K (iti K) Kx to prove that (I P )Kx exists, and that (I P ) X 0 + sup λ (λi K) X 0. λ>0 Moreover, we have (I P ) Kx = Kx, for x belonging to the domain of K, and consequently (I P ) y = y, if y belongs to the closure of R(K). The latter follows because for x belonging to the domain of K we have K (iti K) Kx Kx = itk (iti K) x. (44) The equality in (44) in conjunction with the boundedness of the operator-valued function, restricted X 0, t K (iti K) = I it (iti K), t R \ {0}, yields the existence of the second limit in assertion (v). Next we fix x = (I P ) x + P x X, and we consider e tk x = e tk (I P ) x + e tk P x. Since by (ii) (and the Banach Steinhaus theorem) the expression sup t>0 tke tk X0 is finite, we get lim t e tk y = 0, y R(K). Since, also by (ii), sup t>0 e tk X0 <, we obtain lim t e tk (I P ) x = 0, because the range of K is dense in the range of I P. In addition, we have e tk P x P x = K t 0 esk P x ds =
56 N. KATILOVA P K t 0 esk xds = 0, for all t > 0. Here we used the fact that (I P ) K = K on the domain of K. These observations show that lim t e tk x = P x for all x X 0. Finally (i) also implies that lim λ (λi Reλ>0, λ K) x = x (45) for all x in the domain of K restricted to X 0. Since, by assumption, the domain of K is dense in X, the same is true for its restruction to X 0. Again using (i) implies the equality in (45) for all x X 0. (iv) = (v). This implication is trivial. (v) = (i). Fix x X. From (v) together with the continuity of the function λ (λi K) x, Reλ 0, λ 0, shows that sup Reλ>0 λ (λi K) x < for all x X 0. The theorem of Banach Steinhaus then implies (i). Finally, we give some remarks. We confine the obtained theorems to the case of Markov and Kolmogorov matrices. Since we did not find proofs of the results in Corollary 3.3 and Corollary 3.4 in the literature, we include them here. Remark 3.. Theorem 3. may be useful for Markov matrices M, whereas Theorem 3.6 is more adapted to Kolmogorov matrices K. If M is a Markov matrix, then sup M n =, provided R d is endowed with the supremum-norm. If K is a Kolmogorov matrix, and R d is endowed with the supremum-norm, then sup e tk =. n N t>0 Remark 3.2. A matrix M is analytic in the sense of Definition 3.5, provided the Jordan block corresponding to the eigenvalue has vanishing side diagonals, and the other eigenvalues of M belong to the open unit disc. A matrix K generates a bounded analytic semigroup provided its Jordan block corresponding to the eigenvalue 0 also possesses zero side diagonals. The other eigenvalues are located in the open left half plane. Corollary 3.3. Let the space X be finite dimensional and let M be a Markov matrix. Then M is analytic (in the sense of Definition 3.5 and Blunck [2]), and the Kolmogorov matrix K := M I generates a bounded analytic semigroup (in the sense of Definition 3.0). In fact more is true. From the hypotheses it follows that, for some r >, the expression sup λ <r λ (I λm) is finite. It also follows that, for some ω > 0 the quantity sup Reλ> ω λ (λi K) is finite. Proof. Since M is Markov, it follows that sup n N M n <. With K = M I it also follows that sup t>0 e tk <. Since (λi K) = e λt e tk dt, λ > 0, we 0 get sup λ>0 λ (λi K) <. Using the Jordan decomposition theorem we see that K may be written in the form K = m j=0 (α jp j + N j ). The eigenvalues α j, j m, belong to the open left half plane {λ C : Iλ < 0}, and α 0 = 0; the operators α j P j +N j, 0 j m, commute; the operators P j, 0 j m, are projection operators which commute, and which satisfy P j P k = 0, j k. Finally the operators N j, 0 j m, are nilpotent in the sense that N dj j = 0 for appropriate positive
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 57 integers d j, 0 j d; they also satisfy P j N k = N k P j = δ jk N j, 0 j, k m. (Of course δ jk is the Kronecker symbol.) Since λp 0 (λi K) P 0 = d 0 j=0 λ N j j 0 P 0 we see that, necessarily, N 0 = 0, if lim sup λ 0 λ (λi K) <. But, if N0 = 0, then sup Reλ> ω λ (λi K) < for max {Reαj : j m} < ω < 0. Corollary 3.4. Let M be a finite dimensional Markov matrix with the property that sup n (n + ) M n (M I) < and let K be a finite dimensional Kolmogorov matrix, with the property that sup t>0 tke tk <. Then the following assertions are true: (i) on the range of I M the sequence M n tends to zero exponentially fast; (ii) on the range of K the family e tk goes to zero exponentially fast. Although the result in Corollary 3.4 is well known, a proof is included, because similar arguments show the corresponding result for analytic operators and analytic semigroups in Banach spaces; see Theorem 3.6 below. Proof. Let P be the projection on the null space of I M as described in Theorem 3.. Fix x X. In order to prove assertion (i) we use the representation M n x P x = M n (I P ) x = 2πi λ n+ (I P ) (I λm) x dλ, (46) λ =r where r is some real number which is strictly larger than. Consequently we have: M n x P x r n sup (I P ) (I λm) x. λ =r Next let P the projection as described in Theorem 3.6. In order to prove assertion (ii), we use the representation: e tk x P x = e tk (I P ) x = 2πi (integration by parts) = 2πit ω0+i ω 0 i ω0+i ω 0 i e tλ (I P ) (λi K) 2 x dλ e tλ (I P ) (λi K) x dλ = 2πt e ω0t e itξ (I P ) (( ω 0 + iξ) I K) 2 x dλ, (47) where ω 0 is some strictly positive real number. Consequently e tk x P x e ω0t sup λ 2 (I P ) (λi K) 2 x. ω 0 t Reλ= ω 0 The above representations are justified by the results in Corollary 3.3. Remark 3.5. We notice that the operator (I P ) M n may be written in the form: (I P ) M n = (I M) 2πi λ n (I λm) dλ. (48) λ λ =r
58 N. KATILOVA We also notice the following equalities: (I P ) e tk = 2πi K = 2πit K ω+i ω i ω+i ω i e tλ (λi K) λ dλ { } e tλ (λi K) 2 λ + (λi K) λ 2 dλ = { 2πt K e t( ω+iξ) (( ω + iξ) I K) 2 ω + iξ + (( ω + iξ) I K) ( ω + iξ) 2 = { } e t( ω+iξ) ( ω + iξ) 2 (( ω + iξ)i K) 2 I 2πt ( ω + iξ) 2 dξ. (49) The equalities in (48) and (49) are true whenever M is an analytic operator with an isolated point of the spectrum of M, and K generates a bounded analytic semigroup with 0 an isolated point of the spectrum of K. If we replace formula (46) with formula (48), and formula (47) with (49) we obtain a proof of the following result for analytic operators and analytic semigroups in a Banach space. Theorem 3.6. Let X be a Banach space, and let M be an analytic operator such that sup n N M n <. In addition let K be the generator of a bounded analytic semigroup. Then the following assertions are true: (a) If is an isolated point of the spectrum of M, then X = R (M I) + N (M I), and there exists a finite constant C and a real number r > such that M n (I P ) C, for all n N. (50) rn Here P is the Dunford projection, along the range of M I, on the eigen space of M corresponding to. (b) If 0 is an isolated point of the spectrum of K, then X = R(K) + N(K), and there exists a finite constant C and a constant ω > 0 such that: t e tk (I P ) Ce ωt, for all t > 0. (5) Here P is the Dunford projection on the zero space of K along the closure of the range of K. Corollary 3.7. Let (S, B) be a measurable space, where S is a topological Hausdorff space with Borel field. Let m : B S [0, ] be a sub-probability kernel; i.e. for every x S, the mapping B m (B, x) is supposed to be a sub-probability Borel measure. Consider the linear mapping M : µ m (, x) dµ(x) from the space of all complex measures M b (S, B) to itself. Suppose that the operator M is } dξ
ON MARKOV AND KOLMOGOROV MATRICES AND ANALYTIC OPERATORS 59 analytic and has as eigen-value of multiplicity. Then the operator M is ergodic in the sense that for every non negative probability measure µ, the sequence M n µ converges exponentially fast to the equilibrium measure, as n N converges to. Corollary 3.8. Let (S, B) be a measurable space, where S is a topological Hausdorff space with Borel field. Let p : [0, ) B S [0, ] be a sub Markov family of sub probability kernel; i.e. for every x S, and for every t 0, the mapping B p (t, B, x) is supposed to be a sub-probability Borel measure. Suppose that the mapping p(t, B, x) satisfies the Chapman Kolmogorov identity: p (s, B, x) p (t, dx, y) = p (s + t, B, y). Put e tk µ(b) = p (t, B, x) dµ(x), t 0, B B. Let the semigroup t e tk be analytic. Suppose that 0 is an isolated eigen value of its generator K with multiplicity. Then the semigroup e tk, t 0, is ergodic in the sense that, for every probability measure µ the family e tk µ converges exponentially fast to the equilibrium measure, as t tends to. 4. Conclusions In this article, we consider the relationship between matrices, which satisfy the maximum principle, the sub Markov property, and those which preserve positivity (Proposition 2.). These results allow us to investigate the connection between the matrices with the sub Markov property and the ones with the sub Kolmogorov property (Theorem 2.3). In Proposition 2.4 these relationships are employed for pure Kolmogorov and Markov matrices. We also present a Markov matrix M as an analytic operator (Theorem 3.) and a Kolmogorov matrix K as a generator of a bounded analytic semigroup (Theorem 3.6). We do this by providing conditions on the operator K which guarantee this analyticity and we establish the long time behavior of the corresponding semigroups. We indicate some new properties of these matrices as well. The obtained results yield e.g. Corollary 3.4 about the range of Markov and Kolmogorov matrices. The paper is concluded with similar properties for operators in a Banach space: see Theorem 3.6 and its Corollary 3.7. The proofs rely heavily on Dunford like projections. References. S. Blunck, Analyticity and discrete maximal regularity on L p spaces, J. Funct. Anal. 83 No. (200), 2 230. 2. S. Blunck, Some remarks on operator norm convergence of the Trotter product formula on Banach spaces, J. Funct. Anal. 95 No. 2 (2002), 350 370. 3. N. Katilova, Markov chains: ergodicity in time discrete cases, Rocky Mountain Journal of Mathematics (USA), to appear 2004. 4. Y. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt s type condition, Studia Mathematica, 34 No. 2 (999), 53 67. 5. B. Nagy and J. Zemanek, A resolvent condition implying power boundedness, Studia Mathematica, 34 No. 2 (999) 43 5.
60 N. KATILOVA 6. J.A. Van Casteren, Generators of Strongly Continuous Semigroups, Pitman Advanced Publishing Program, Research Notes in Mathematics 5, 985. 7. J. Van Casteren, Progress in Nonlinear Differential Equations and their Applications, Birkhuser Verlag Bazel/Switzerland, Vol. 42, 2000. N. Katilova Department of Mathematics and Computer Science University of Antwerp (UA) Middelheimlaan 2020 Antwerp BELGIUM