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Владикавказский математический журнал 2013, Том 15, Выпуск 1, С. 41-50
УДК 517.98
ON POSITIVE INVERTIBILITY AND SPLITTINGS OF OPERATORS IN ORDERED BANACH SPACES
K. C. Sivakumar1, M. R. Weber
The authors dedicate the paper to Prof. A. G. Kusraev on the occasion of his sixtieth birthday
The positive invertibility of operators between Banach spaces, ordered by special closed cones, is characterized by the existence of splittings for the operators into the difference of two operators with appropriate spectral properties. Some results, up to now known only for matrices, are generalized to operators and to order intervals of operators.
Mathematics Subject Classification (2000): 46B40, 47B60, 47B65, 46A40, 47A05.
Key words: ordered Banach spaces, cones in ordered spaces, positively invertible operators, splitting of operators, intervals of operators.
1. Introduction
Let (X, X+, ||-||) and (Y, Y+, ||-||) be ordered normed spaces, where the order is being introduced by means of the cones X+ and Y+, respectively. Briefly, we will write X and Y instead of (X,X+, ||-||) and Y for (Y, Y+, ||-||), respectively. Denote by L(X, Y) the vector space of all linear continous operators A: X ^ Y. We shall write L(X) for L(X, X). An operator A e L(X, Y) is said to be positive if A(X+) C Y+, and is said to be positively invertible (or, inverse-positive) if it is invertible and A-1(Y+) C X+. We write for these properties simply A ^ 0 and A-1 ^ 0, respectively.
Many iterative methods for solving matrix equations Ax = b in numerical analysis are based on splittings of the corresponding matrix A, see [22]. As usual, a splitting of the matrix A into the difference of two matrices, say A = U — V with an invertible matrix U, is used to associate to A the iteration method
Uxk+1 = Vxk + b, k e No, where N0 = N U {0}. This procedure can be rewritten as
xk+1 = u-1 Vxk + U-1b, k e No.
© 2013 Sivakumar K. C., Weber M. R.
1 This author thanks for funds and hospitality during his stay in summer 2010 at the Technical University Dresden.
It is well known that several questions concerning the matrix A and the iteration method, e. g. the convergence rate of iteration methods, can be answered by means of properties of the operators involved in the splitting (see [21, 22]).
Positively invertible operators are also of interest in connection with the existence of a positive solution for equations
Ax = y, y £ Y+,
where A: X ^ Y is an operator between the ordered normed spaces X and Y (see for e.g. [9, 11, 13, 14, 20, 27]).
The positive invertibility of matrices was chacterized in [18] by what later has been called B-splittings. This result has been generalized to operators in ordered normed spaces in [2527]. In the recent work [10] the positivity of generalized inverse matrices, such as Moore-Penrose and group inverses, is studied by means of adapted B-splittings and, conditions are given when such B-splittings exist.
The objective of the present article is to generalize certain results, up to now known only for matrices, to the cases of operators and operator intervals in ordered normed spaces.
2. Preliminaries
Let (X, X+, ||-||) be an ordered normed space, where throughout the cone X+ is understood to be closed. We need some additional properties of cones2 in ordered normed spaces, which characterize further relations or the compatibility between the order and the norm in such spaces. The cone X+ of an ordered vector space (X, X+) is reproducing (or generating), if any x £ X has a representation x = u — v, where u, v £ X+. The cone X+ of the ordered normed space (X, X+, ||-||) is said to be normal, if there exists a constant N (the constant of normality) such that x,y £ X and 0 ^ x ^ y imply ||x|| ^ N ||y||. In this case the norm is called semi-monotone. A cone X+ is said to be regular, if each monotone increasing sequence of elements of X+ that is order bounded (from above), is norm-Cauchy. In a Hilbert space any closed normal cone is regular (see [17]). If a cone in an ordered normed space is regular and possesses interior points then it is normal and, the norm is order continuous (see [24, Theorem I.5.2]). Each closed regular cone in a Banach space is normal (see [15, Theorem 5.1]).
In the vector space L(X, Y) of all linear continuous operators A: X ^ Y, where X is an ordered normed space and Y stands for the ordered normed space (Y, Y+, ||-||), the subset L+(X, Y) = {A £ L(X, Y): A(X+) c Y+} is, in general, a closed wedge and, under the assumption that the linear set X+ — X+ is dense in X, it is a cone in L(X, Y) (see [23]).
If Y = R1 and Y+ = R+, then L+(X, Y) is denoted by X+ and is called the dual wedge or cone, respectively.
If X = Y (this means also Y+ = X+ and the identity of the norms) then L(X, X) is denoted by L(X) and L+(X, Y) by L+(X).
There are known several conditions for the cone L+(X, Y) to be reproducing, normal and regular (see [2, 3, 5, 24]). We formulate them in the form we will use them later.
Theorem 2.1. 1. If the cones X+ and Y+ are both normal, int(Y+) = 0 and (Y, Y+) is a Dedekind complete vector lattice then the wedge L+(X,Y) is reproducing, i.e., any A £ L(X, Y) can be represented as the difference of two positive operators: A = U — V, where U, V £ L+(X,Y).
2 In several cases the set X+ may be supposed to be a wedge.
2. Let X + = X. For the cone L+(X, Y) to be normal it is necessary and sufficient that Y+ is normal and X+ satisfies the condition: there exist a constant M > 0 such that for each x £ X there are two sequences (un) and (vn) such that un, vn £ X+ with ||un||, ||vn|| ^ M ||x|| and x = lim(un — vn). In particular, if X is a Banach space then the cone L+ (X, Y) is normal if and only if the cone X+ is reproducing and Y+ is normal.
3. If int(X+) = 0 and Y+ is normal and regular, then L+ (X, Y) is normal and regular.
4. Let (X,X+, W'W) and (Y, Y+, be ordered normed spaces. For the wedge L+(X, Y) to have nonempty interior it is necessary and sufficient that the cone X + allows plastering and the cone Y+ has interior points.
Definition 2.2. Let be A £ L(X, Y). An operator T £ L(Y,X) is called the inverse of A if AT = IY and TA = IX, where IX, Iy are the identity operators in X and Y, respectively. T is uniquely defined and denoted by A-1.
If the operator A £ L(X, Y) is invertible then the continuity of A-1 follows either from Banach's theorem (if X and Y are Banach spaces) or, if A-1 ^ 0 is known, from continuity theorems for positive operators3 (see [1, 24]).
For an operator A £ L(X) the spectrum is denoted by a (A) and the spectral radius is defined as r(A) = sup{|A|: A £ a(A)}. For the spectral radius there holds the Gelfand formula
r(A) = lim n|An| = inf VHA^.
n—n
It is well known (see e.g. [12, chapter V]) that r(A) < 1 for an operator A £ L(X) is a sufficient condition for the convergence of the series
I + A + A2 + ... + An + ... (1)
The spectral radius of an invertible operator A is positive. Indeed 1 = r(I) ^ r(A)r(A-1) implies r(A) > 0 and r(A-1) > 0. If an operator A is positively invertible then r(A-1) £ a(A-1) (see [15, Theorem 8.1]). For two operators A, B with — B ^ A ^ B in a space X with a reproducing and normal cone one has r(A) ^ r(B) (see [15, Theorem 16.5]). The last inequality, in particular, holds for positive operators with 0 ^ A ^ B.
The next theorem provides some well known conditions on an operator C in a Banach space X to guarantee the invertibility of the operator I — C in L(X).
Theorem 2.3 [27, Theorem 2.1]. Let X be a Banach space and C: X ^ X be a continuous linear operator on X. Consider the following properties:
(i) the spectral radius of C satisfies r(C) < 1;
(ii) C is quasinilpotent, i. e. lim ||Cn|| = 0;
n—
(iii) there exists the inverse operator (I — C)-1, where
(I — C )-1 = I + C + C2 + ... + Cn + ... (2)
Then there hold the implications: (i) (ii) (iii).
If X = (X,X+, |H|) is an ordered Banach space and the operator C is positive, i.e. C £ L+(X), then from formula (2) it is easy to see, that (i) implies even (I — C)-1 ^ 0.
3 If X is an ordered Banach space such that each positive linear functional on X is continuous and Y is an ordered Banach space with a closed cone then any linear positive operator A: X ^ Y is continuous (Lozanovski's Theorem). The condition for X holds e.g. if int(X+) = 0 or if X+ is closed and reproducing. The norm-completeness of Y can be removed if the cone Y+ in the ordered normed space Y is assumed to be normal. Of course, in case of normed lattices the situation is much simpler: If X is a Banach lattice and Y any normed lattice then each positive operator is continuous.
The implication (iii) ^ (i) holds if the cone X+ in the ordered Banach space has additional properties. Moreover, if the operator C is compact and satisfies r(C) > 0 then the number r(C) turns out to be an eigenvalue of C possessing a positive eigenvector4. More precisely, one has the following result (see [15, Theorem 9.2 and §25]).
Theorem 2.4. Let (X, X+, ||-||) bean ordered Banach space and C: X ^ X a continuous linear operator on X such that C ^ 0.
(i) Then r(C) < 1 implies (I — C)-1 ^ 0 and, if the cone X+ is normal and reproducing, also vice versa, i. e. the existence of (I — C)-1 and (I — C)-1 ^ 0 imply r(C) < 1.
(ii) Let the cone X+ satisfy the condition X+ — X+ = X. If C is compact and r(C) > 0 then r (C) is an eigenvalue of C possessing a positive eigenvector. If additionally (I—C)-1 ^ 0 then r(C) < 1.
< (i): If r(C) < 1 and C ^ 0 then by Theorem 2.3 the operator (I — C)-1 exists and (I — C)-1 ^ 0 immediately follows from (2). The converse statement is Theorem 25.1 of [15].
(ii): The compactness of the positive operator C implies the existence of a positive eigenvector to the eigenvalue r(C), so there exists a nonzero vector xo G X+ such that Cx0 = r(C)x0. If there would be r(C) = 1 then (I — C)x0 = 0 contradicts the invertibility of I — C. Consequently, r(C) = 1 and (I — C)x0 = (1 — r(C))x0 implies 1-r1(-C)x0 = (I — C)-1x0. Since x0 > 0 and (I — C)-1 ^ 0 there must hold r(C) < 1. >
Let again X and Y be ordered normed spaces. For two operators B,C G L(X, Y) with B ^ C the (order) interval [B, C] is defined as the subset of all operators "between" B and C, i. e.
[B, C] = {A G L(X, Y): B ^ A ^ C}.
An operator interval t = [B, C] is said to be invertible (positively invertible), if each operator A G t is invertible (positively invertible).
In the theory of positive operators the following question arises in several situations and is studied by many authors: Assume the "endpoints" of an operator interval have some property (P). Do all operators A G [B,C] share this property too? If 0 ^ A ^ C and the property (P) holds for C then the "domination problem for positive operators" is equivalent to the question: does A satisfy (P)? It turns out that under appropriate conditions on the spaces X and Y the answer for many properties (P) is affirmative. In this regard, the following result is pertinent and will also be used later.
Theorem 2.5 ([14, Theorem 1] and [15, Theorem 25.4]). Let be X = (X,X+, ||-||) an ordered normed space and Y = (Y, Y+, ||-||) an ordered Banach space with a normal cone Y+ that satisfies the condition int(Y+) = 0. Let be B,C: X ^ Y two linear continuous operators, where B ^ C and C is positively invertible. Then B is positively invertible if and only if B (X+) n int(Y+) = 0.
3. Splittings and positive invertibility
In what follows X = (X, X+, ||-||) and Y = (Y, Y+, ||-||) will be two ordered normed spaces and A G L(X, Y) is an arbitrary operator represented as a difference of two operators, i.e., A = U — V, where U, V G L(X, Y). We consider the following splittings5 of the operator A.
4 Such eigenvalues are called Perron-Frobenius-eigenvalues.
5 In former papers, e.g. in [27], splittings were called decompositions.
U ^ 0 V ^ 0 U^ 0 U-1V ^ 0 VU-1 ^ 0 other condition
positive x x
regular x x
positively regular x x x
weak regular x x
B-splitting x x x (B)
Ax > 0 1
where condition (B) is the following implication: > o | x ^ 0-
These splittings are generalizations (to the infinite-dimensional case) of the corresponding splittings known for square matrices, where X = Y = Rn (see [16, 18, 22, 27]).
As was mentioned in [27], the condition VU-1 ^ 0 is equivalent to the implication Ux ^ 0 Vx ^ 0.
If an operator A has a positive splitting A = U — V such that U-1 exists, then it can be represented as
A = U — V = (Iy — VU-1) U, with U, V ^ 0. (3)
Our first main result to follow is to prove a generalization of a result of Varga ([22, Theorem 3.37]) to the operator case.
Theorem 3.1. Let be X an ordered Banach space and Y an ordered normed space such that the cone L+(Y, X) is regular6. Suppose an operator A e L(X, Y) has a weak regular splitting A = U — V, i.e., U-1 ^ 0 and U-1V ^ 0. Then the operator A is positively invertible if and only if r(U-1V) < 1.
< Let A = U — V be the given weak regular splitting of A. Set C = U-1V: X ^ X. Then C ^ 0. Also
A = U(IX — U-1V) = U(IX — C), so that U-1A = U-1(U — V) = IX — C. (4)
If r(C) < 1 then by Theorem 2.4 (i) the operators Ix — C and A are positively invertible,
and A-1 = (Ix — C)-1U-1.
Conversely, if A-1 ^ 0 then the representation of U-1A in (4) gives U-1 = (Ix — C)A-1
and (Ix + C)U-1 = (Ix + C)(Ix — C)A-1 = (Ix — C2)A-1. For any integer k define
Bk = (Ix + C + C2 + ■ ■ ■ Ck)U-1. Then Bk > 0, Bk = (Ix — Ck+1)A-1 and (Bk+1 — Bk)U =
Ck+1. By taking into consideration Ck+1A-1 ^ 0, it then follows that 0 ^ Bk ^ A-1.
Consequently, (Bk) is an order bounded increasing sequence in the ordered Banach space
L(Y,X) with a regular cone L+(Y, X). So, the sequence (Bk) converges in L(Y,X). Due to
||Bk+1U — BkU|| ^ ||Bk+1 — Bk|| ||U|| the sequence (BkU) converges in L(X). In particular,
||Bk+1U — BkU|| = ||Ck+1y —^ 0. Since 11CkM < 1 for sufficiently large k, the Gelfand
k
formula shows that r(C) = infn ^||Cn| < V||Ck|| < 1. >
The next two theorems are generalizations to the case Y = X of some results of [27]. Since the proofs are similar, they are omitted. In particular, the first could be viewed as an extension of well-known characterizations of M-matrices (see [8, 22]).
Theorem 3.2 ([27, Theorem 3.4] and [25, Theorem 3 generalized]). Let X = (X,X+, ||-||) and Y = (Y, Y+, ||-||) be ordered Banach spaces, where X+ and Y+ are normal cones and Y+
6 According to Theorem 2.1 this holds, e.g. if int(Y+) = 0 and X+ is normal and regular.
satisfies the condition int(Y+) = 0. Let A: X ^ Y be a linear continuous operator. Consider the conditions:
(i) A is positively invertible,
(ii) Y+ c A(X+),
(iii) there exists some x0 G X+ such that Ax0 G int(Y+).
Then we have the implications (i) ^ (ii) ^ (iii).
Consider the following condition under the assumption that A possesses a B-splitting:
(iv) r(VU-1) < 1.
Then ^ (iii) ^ (iv) ^ (i), i. e., in this case all conditions are equivalent.
The question of existence of a B-splitting is also settled in the affirmative in the next result.
Theorem 3.3 ([27, Theorem 3.7 generalized]). Let X = (X,X+, ||-||) and Y = (Y,Y+, ||-||) be ordered Banach spaces, where the cone X+ allows plastering and Y+ satisfies the condition int(Y+) = 0 .If the operator A G L(X, Y) is positively invertible then A possesses a B-splitting A = U — V such that r(VU-1) < 1.
Remark. 1. If the operator A is invertible then (ii) is a sufficient condition for the positivity of its inverse. Indeed, apply A-1 to the inclusion (ii).
2. Only the proof of the implication (iv) ^ (i) requires the condition (B) of a B-splitting.
3. If A: X ^ Y is an invertible operator then A-1 ^ 0 if and only if Y+ c A(X+). Indeed, A-1 ^ 0 means A-1(Y+) c X+. After applying A the required inclusion follows. If A-1 exists, the application of the operator A-1 to Y+ c A(X+) yields A-1(Y+) c X+, i.e. A-1 ^ 0.
In an ordered Hilbert space for a selfadjoint operator with a positive regular splitting the first three conditions of Theorem 3.2 are equivalent to another spectral condition (vi):
Theorem 3.4. Let H = (H, H+, ||-||) be an ordered Hilbert spaces, where H+ is a closed, normal cone such that int(H+) = 0. Let A: H ^ H be a linear continuous selfadjoint operator. Consider the conditions:
(i) A is positively invertible,
(ii) H+ c A(H+),
(iii) there exists some x0 G H+ such that Ax0 G int(H+).
Then we have the implications (i) ^ (ii) ^ (iii). Consider the following condition under the assumption that A possesses a positive regular splitting: A = U — V with U ^ 0, V ^ 0, U-1 ^ 0
(vi) r(U-1V) < 1.
Then (iii) ^ (vi) ^ (i), i. e., in this case all four conditions are equivalent.
< The proof of the implications (i) ^ (ii) ^ (iii) follows as a particular case of the corresponding parts of Theorem 3.2. We prove only the implications (iii) ^ (vi) and (vi) ^ (i). Let A = U — V be a splitting of A such that U ^ 0, V ^ 0, U-1 ^ 0. Set C = U-1V. Then C ^ 0 and, similar to (4), A = U(I — C). Moreover, A* = (l — C*)U*.
(iii) ^ (vi): From C ^ 0 it follows that C* ^ 0. So, I — C* ^ I. By assumption there is a vector x0 G H+ with Ax0 = A*x0 G int(H+), i.e. (I — C*)U*x0 G int(H+). Since U*x0 G H+, the positive invertibility of I — C* is now guaranteed by Theorem 2.5. Using (i) of Theorem 2.4, it follows that r(C*) < 1, i.e. r(C) < 1.
(vi) ^ (i): Since any positively regular splitting is weak regular, the proof follows from Theorem 3.1 provided the cone L+(H) is regular. The last condition is guaranteed by Theorem 2.1 (3), since the cone H+ is closed and normal, and therefore, by [17] it is regular. >
Remark. Due to the previous Remark 2 under the assumptions of the Theorem the condition (iv) r(VU-1) < 1 also holds.
4. On operator intervals
For operator intervals the following theorem is well known.
Theorem 4.1 ([13, Theorem 3], [14, Theorem 4]). Let (X,X+, ||-||) = X be ordered normed space and (Y, Y+, ||-||) = Y an ordered Banach space, with closed cones X+ and Y+, respectively. Let the cone Y+ be generating and normal. Let B,C e L(X, Y) such that B ^ C. Then each operator A e [B, C] is positively invertible if and only if the operators B, C are both positively invertible.
This result was reproved for interval matrices later in [19], where the condition B-1 ^ 0 has been equivalently replaced either by a condition on the spectral radius r(C-1(C — B)) or the condition that all matrices of the interval [B, C] are invertible.
We prove now our second main result, the operator version of Rohn's result, namely
Theorem 4.2. Let (X, X+, ||-||) and (Y, Y+, ||-||) be ordered normed spaces, where (Y, Y+) is Dedekind complete. Assume the cones X+ and Y+ satisfy the conditions int(X+) = 0, X+ is normal, int(Y+) = 0, Y+ is normal and regular. Let be t = [B,C] the operator interval defined by two given operators B,C e L(X, Y) with B ^ C. Then the following assertions are equivalent
(a) t is positively invertible,
(b) C-1 ^ 0 and B-1 ^ 0,
(c) C-1 ^ 0 and r(C-1 (C — B)) < 1.
< (a) (b) is trivial.
(b) (c): Put U = C and V = U — B. Then obviously, U-1V ^ 0. Notice that B = U — V is a weak regular splitting. By Theorem 3.1 the inequality B-1 ^ 0 implies r(U-1V) = r(C-1(C — B)) < 1.
(c) (a): Let A be an arbitrary operator in t represented as A = U — V, where U = C and V = C — A. Then U)-1 = C-1 ^ 0 and V ^ 0 so that U-1V ^ 0. We also have r(U-1V) = r(C-1(C — A)) < r(C-1(C — B)) < 1. By Theorem 3.1, it now follows that A is positively invertible. >
Corollary 4.3. Denote by (d) the statement C-1 ^ 0 and assume that t is invertible, i. e., A-1 exists for each A e t. Under the conditions of Theorem 4.2 suppose that the operator C — B is compact. Then (d) (c) and, so all conditions (a)-(d) are equivalent.
< We use the same argument as in the finite dimensional setting (see [19]). If the identity operator in X is denoted by I then C-1(C — B) = I — C-1B. Suppose the contrary, i.e. r = r(I — C-1B) ^ 1. Since the operator I — C-1B is positive and compact (and obviously, r > 0) by (ii) of Theorem 2.4 we have that r is an eigenvalue of the operator ( — C-1B. Consequently, for some nonzero vector x0 one has then C-1(C — B)x0 = rx0, i.e. ((1 — 1 )C + 1 B)x0 = 0. Set A = r and D = AB + (1 — A)C. Then 0 < A ^ 1 and hence D is a convex combination of the operators B and C. So, D e t. But Dx0 = 0 is a contradiction, as t is invertible. >
Corollary 4.4. Under the conditions of Theorem 4.2 suppose that one of the conditions (a)-(c) holds. Then for A e t we have the following representation of its inverse:
A-1 = ( E(C-1(C — A))k] C-1.
It follows that A-1 ^ 0.
< Denote the identity operator in X by I and represent A as A = C(I — C-1(C — A)). Then, by the hypotheses and by what has been mentioned in § 2 for the spectral radii of positive operators, one has r(C-1(C — A)) ^ r(C-1(C — B)) < 1. Therefore,
(I — C-1(C — A))-1 ^ 0 and (I — C-1(C — A))-1 = (C-1(C — A))k.
k=0
It follows immediately that A-1 = ( (C-1(C - A))fc)C-1 ^ 0. >
Remark. 1. From (b) of the theorem also follows that r(Iy — BC-1) < 1. Indeed,8 notice that the positive operator CB-1 = (C — B)B-1 + Iy is the inverse of BC-1. Define the operator A by A = Iy — BC-1 then A = (C — B)C-1, what yields A ^ 0. Since Iy — A = BC-1, by (i) of Theorem 2.4 the positive invertibility of Iy — A implies r(A) < 1, i.e. r (Iy — BC-1) < 1.
2. If for some operator B : X ^ Y there exist a positively invertible operator C such that B ^ C, the interval [B,C] is invertible and the operator C — B is compact, then A-1 ^ 0 for any operator A G [B,C]. This immediately follows from the implications (d) (c)
(a).
3. Other conditions for a positive operator to possess non-zero positive eigenvalues are studied in [4, 6, 7]. They also might be used to prove sufficient conditions for the positive invertibility of appropriated operator intervals.
4. If B < C and C-i ^ 0, B-i > 0 then C-i < B
1-i
)-i 1-1,
Indeed, B 1 ^ 0 implies
BB-1 < CB-1, i.e. Iy < CB-1. So C-1 ^ 0 gives C-1 < C-1CB-1, i.e. C-1 < B
In general, even for matrices, the conditions B ^ C, C-1 ^ 0 and the existence of B do not imply C-1 ^ B-1. Consider
-i
B =
0 -1 -1 0
B-i =
0 -1 10
and C =
C-i =
2 1 1 2
Then B < C, C-1 ^ 0 and B-1 < C-1. The result C-1 < B-1 holds for invertible M-matrices B, C with B < C (see [22, §3.5]).
5. The theorem remains true if Y is a Banach space and Y+ is assumed to be a closed regular cone, since as was already mentioned, in this case the cone Y+ is normal.
2
3
3
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Received April 30, 2012.
Sivakumar Koratti Chengalrayan Indian Institute of Technology Madras Department of Mathematics, Prof. 600 036, Chennai, India E-mail: kcskumar@iitm.ac.in
Weber Martin Richard Technische Universität Dresden Fachrichtung Mathematik, Prof. 01069 Dresden, Germany E-mail: martin.weber@tu-dresden.de
О ПОЛОЖИТЕЛЬНОЙ ОБРАТИМОСТИ И РАЗЛОЖЕНИИ ОПЕРАТОРОВ В УПОРЯДОЧЕННЫХ ПРОСТРАНСТВАХ БАНАХА
Сивакумар К. Ч., Вебер М. Р.
Положительная обратимость операторов, действующих между упорядоченными пространствами Банаха, характеризуется с помощью разложений операторов в разность двух операторов с подходящими спектральными свойствами. Некоторые результаты, известные до сих пор только для матриц, обобщаются на операторный случай и на операторные интервалы.
Ключевые слова: упорядоченные пространства Банаха, конуса в упорядоченных пространствах, положительная обратимость операторов, разложение операторов, операторные интервалы.
