Banach Algebras Associated with Linear Operator Pencils

ISSN 0001-4346, Mathematical Notes, 2009, Vol. 86, No. 3, pp. 361–367. © Pleiades Publishing, Ltd., 2009. Original Russian Text © I. V. Kurbatova, 2009, published in Matematicheskie Zametki, 2009, Vol. 86, No. 3, pp. 394–401.

Banach Algebras Associated with Linear Operator Pencils I. V. Kurbatova* Voronezh State University Received January 15, 2008; in final form, October 4, 2008

Abstract—A direct relationship between the theory of pseudoresolvents and the spectral theory of linear operator pencils is established. DOI: 10.1134/S0001434609090090 Key words: differential equation not solved with respect to derivatives, operator pencil, resolvent, spectrum, pseudoresolvent, maximal pseudoresolvent, Banach algebra.

Differential equations of the form F x˙ = Gx + f (t) not solved with respect to derivatives arise in many applications; see, e.g., [1]–[5]. Here F, G : X → Y are (possibly, unbounded) linear operators, and X and Y are Banach spaces, For a differential equation x˙ = Gx + f (t) solved with respect to the derivative, finding a general solution reduces to constructing the operator exponential eGt ; a natural method for constructing and studying such an operator exponential is based on the spectral properties of the operator G. The transfer of this approach to equations not solved with respect to the derivative was considered by many authors [5]–[15]. The first step in its implementation consists in replacing the resolvent (λ1 − G)−1 of the operator G by the resolvent (λF − G)−1 of the pencil λ → λF − G. Unfortunately, as a rule, the domain Y of the pencil resolvent (λF − G)−1 does not coincide with its range X. For this reason, consideration of the spectral properties of pencils usually begins by passing to the space X or Y (see, e.g., [5], [6], [11]). The objective of this paper is to describe a commutative Banach algebra generated by the pencil λ → λF − G (see Theorem 8), which reduces the study of the properties of the resolvent (λF − G)−1 to a direct application of classical facts from spectral theory [16]–[18]. In Sec. 1, we recall the terminology of the theory of Banach algebras [16]–[18] and basic properties of pseudoresolvents [18]. In Sec. 2, we define a Banach algebra B(F,G) (Y, X∗ ) under F -multiplication (see Theorem 8). The resolvent of the pencil (λF − G)−1 turns out to be a maximal pseudoresolvent with values in this algebra (see Proposition 9). As an application, we prove Theorem 11, which describes functional calculus, and Theorem 12 on spectral decomposition. These theorems imply, in particular, Corollary 14 on the decomposition of X and Y into the direct sum generated by a partition of the extended singular set of the pencil resolvent into two parts, which is the best known spectral fact in the theory of operator pencils [3], [5], [6], [8], [9], [11], [14], [15]. 1. PRELIMINARIES All algebras considered below are assumed to be complex [16]–[18]. If an algebra B contains an element 1 = 1B ∈ B such that A1 = 1A = A for any A ∈ B, then this element 1 is called a unit, and the algebra B is said to be unital. An algebra can have at most one unit. An algebra is said to be commutative if AB = BA for any A and B. If an algebra B is a Banach space and AB ≤ A · B, then B is called a Banach algebra. If, in addition, B is unital and 1 = 1, then B is a unital Banach algebra. The simplest example of a Banach algebra is the algebra B(X) of bounded linear operators acting on a Banach space X. *

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Let B be a unital algebra. The inverse of A ∈ B is defined as an element A−1 ∈ B for which AA−1 = A−1 A = 1. For a unital algebra B and its element A ∈ B, the set of λ ∈ C for which the element λ1 − A has no inverse is called the spectrum of A and denoted by σ(A) or σB (A). Its complement ρ(A) = C \ σ(A) is the resolvent set of A. The function (the family) λ ∈ ρ(A) Rλ = (λ1 − A)−1 , is called the resolvent of A. The spectrum of any nonzero element of a unital Banach algebra is a nonempty compact subset of C. Proposition 1 ([18, Theorem 4.1.8]). If B is a unital algebra, then the resolvent Rλ of any element A ∈ B satisfies Hilbert’s identity λ, μ ∈ ρ(A). (1) Rλ − Rμ = −(λ − μ)Rλ Rμ ,  = C ⊕ B under the coordinatewise linear Let B be an algebra without unit. Obviously, the set B operations and the multiplication (α, A)(β, B) = (αβ, αB + βA + AB) is an algebra with the unit  is referred to as the algebra B 1 = (1, 0). The element (α, A) is denoted by α1 + A. The algebra B with adjoint unit. If the algebra B is Banach, then we set α1 + A = |α| + A; obviously, in this  is Banach as well. The spectrum (resolvent) of an element in a nonunital algebra case, the algebra B is defined as the spectrum (resolvent) of this element in the algebra with adjoint unit. If B is unital, then  with adjoint unit we understand the algebra B itself. by the algebra B Let A and B be algebras. A map ϕ : A → B is called an algebra morphism [16] if ϕ(A + B) = ϕ(A) + ϕ(B),

ϕ(αA) = αϕ(A),

ϕ(AB) = ϕ(A)ϕ(B).

If A and B are unital and ϕ(1A ) = 1B , then ϕ is said to be a unital algebra morphism. Suppose that A and B are algebras and A has no unit. If ϕ : A → B is an algebra morphism, then,  →B  is the map obviously, the unique extension of ϕ to a unital algebra morphism ϕ : A ϕ(α1  + A) = α1 + ϕ(A). Let B be a Banach algebra, and let G ⊆ C be a nonempty set. A pseudoresolvent (on G with values in B) is a function (family) λ → Rλ defined on G, taking values in B, and satisfying Hilbert’s identity (1) [18, Chap. 5, Sec. 2, p. 201 (Russian transl.)]. A pseudoresolvent is said to be maximal [6] if it has no extension to a larger set satisfying identity (1). Any pseudoresolvent can be uniquely extended to a maximal pseudoresolvent (see Theorem 3). The domain ρ(R( · ) ) of this maximal pseudoresolvent is called the regular set of the initial pseudoresolvent, and the complement σ(R( · ) ) to ρ(R( · ) ) is its singular set. Hilbert’s identity (1) can be written in the equivalent form Rλ + (λ − μ)Rλ Rμ = Rμ or in the form Rλ (1 + (λ − μ)Rμ ) = Rμ (if the given algebra is nonunital, then the algebra with adjoint unit is considered). Proposition 2 ([18, Corollary 1 of Theorem 5.8.4]). If two commuting elements Rλ , Rμ ∈ B satisfy (1), then the element  1 + (λ − μ)Rμ ∈ B is invertible. Theorem 3 ([18, Theorem 5.8.6]). Any pseudoresolvent admits a unique extension to a maximal pseudoresolvent. The domain of this maximal pseudoresolvent is the set of all λ ∈ C for which  The extension has the form the element 1 + (λ − μ)Rμ is invertible in B. Rλ = Rμ (1 + (λ − μ)Rμ )−1 . Corollary 4 ([18, Theorem 5.8.2], [13, Chap. 6, Sec. 1]). The domain of any maximal pseudoresolvent is an open set, and such a pseudoresolvent itself is an analytic function with values in B. MATHEMATICAL NOTES

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2. F -ALGEBRAS Given complex linear spaces X and Y , linear subspaces XF , XG ⊆ X, and linear operators F : XF → Y and G : XG → Y , the (operator) pencil is defined as a function λ → λF − G : X∗ → Y,

λ ∈ C,

where X∗ = XF ∩ XG [5], [6], [10]. In particular, 0F − G denotes the operator −G : X∗ → Y . It follows from Proposition 7 below that the case of bounded operators F, G : X∗ → Y can be considered general. The resolvent set of such a pencil is the set ρ(F, G) consisting of those λ ∈ C for which the operator λF − G : X∗ → Y is invertible, and its resolvent is the function (the family) Rλ = (λF − G)−1 : Y → X∗ ,

λ ∈ ρ(F, G).

The complement σ(F, G) to ρ(F, G) is called the spectrum of the pencil. Proposition 5. The resolvent of a pencil satisfies Hilbert’s F -identity Rλ − Rμ = −(λ − μ)Rλ F Rμ ,

λ, μ ∈ ρ(F, G).

(2)

Proof. Indeed, for λ, μ ∈ ρ(F, G), we have Rλ = (λF − G)−1 = (λF − G)−1 (μF − G)(μF − G)−1   = (λF − G)−1 λF − G + (μ − λ)F (μF − G)−1 = (λF − G)−1 (λF − G)(μF − G)−1 + (λF − G)−1 (μ − λ)F (μF − G)−1 = (μF − G)−1 + (μ − λ)(λF − G)−1 F (μF − G)−1 = Rμ − (λ − μ)Rλ F Rμ . Suppose that the following hypothesis holds. Conjecture. The space Y is Banach (on X, no norm is defined in advance), and ρ(F, G) contains at least two points λ = μ for which the operators (λF − G)(μF − G)−1 : Y → Y,

(μF − G)(λF − G)−1 : Y → Y

are bounded. Proposition 6. Under the above conjecture, the isomorphisms λF − G, μF − G : X∗ → Y generate equivalent norms on X∗ . In what follows, we assume X∗ to be endowed with one of the equivalent norms generated by the isomorphisms λF − G and μF − G. Proposition 7. If λ, μ ∈ ρ(F, G), λ = μ, and the operators (λF − G)(μF − G)−1

and

(μF − G)(λF − G)−1

are bounded, then the operators F : X∗ → Y and G : X∗ → Y are bounded as well. Thus, for any ν ∈ ρ(F, G), the operator νF − G : X∗ → Y generates a norm on X∗ equivalent to the norms generated by λF − G and μF − G. Proof. By assumption, the operators (λF − G) : X∗ → Y and (μF − G) : X∗ → Y are bounded. Therefore, so is the operator (λF − G) − (μF − G) = λF − μF. Thus, the operator F : X∗ → Y is bounded. But this implies the boundedness of the operator G = λF − (λF − G) : X∗ → Y and of the operator νF − G : X∗ → Y . The second assertion follows from Banach’s inverse operator theorem. MATHEMATICAL NOTES

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We emphasize that the space X may be Banach, and the operators F : XF → Y and G : XG → Y may be unbounded. According to Proposition 7, under the assumption made above, even unbounded operators F : XF → Y and G : XG → Y generate bounded operators F, G : X∗ → Y . This reduces the case of unbounded operators to that of bounded operators. Let B(Y, X∗ ) denote the Banach space of all bounded linear operators A : Y → X∗ , and let B(F,G) (Y, X∗ ) be the closure in B(Y, X∗ ) of the linear span of the operators Rλ with λ ∈ ρ(F, G). On B(F,G) (Y, X∗ ), we define the F -multiplication by setting A  B = AF B. Theorem 8. The space B(F,G) (Y, X∗ ) is a commutative Banach algebra under F -multiplication. This algebra is unital if and only if the operator F : X∗ → Y is invertible; in this case, the unit is F −1 . If the algebra B(F,G) (Y, X∗ ) is unital, then an algebraic norm (i.e., a norm satisfying the condition A  B ≤ A · B) on this algebra can be defined as   AF = sup A  B : B ≤ 1, B ∈ B(F,G) (Y, X∗ ) . If the algebra B(F,G) (Y, X∗ ) is nonunital, then the function   AF = sup βA + A  B : |β| + B ≤ 1, β ∈ C, B ∈ B(F,G) (Y, X∗ ) can be taken for the algebraic norm. Both these norms are equivalent to the initial norm on the space B(Y, X∗ ). Proof. Let us check that B(F,G) (Y, X∗ ) is closed under F -multiplication. It follows from Hilbert’s identity (2) that the product Rλ Rμ with λ = μ belongs to the linear span of the family of Rλ with λ ∈ ρ(F, G). By continuity, the elements Rλ Rμ with λ = μ belong to the closure of the linear span of the family of Rλ with λ ∈ ρ(F, G). Obviously, F -multiplication is continuous with respect to the initial norm on the space B(Y, X∗ ). It follows that the closure of the linear span of Rλ with λ ∈ ρ(F, G) forms an algebra. Hilbert’s identity (2) also implies the permutability of Rλ and Rμ for λ, μ ∈ ρ(F, G). By continuity, this implies the commutativity of the algebra B(F,G) (Y, X∗ ). Let us check the existence of a unit. If F is invertible, then, obviously, we can take F −1 for 1F . Conversely, suppose that the algebra has a unit 1F . Substituting a surely invertible operator (e.g., one of the resolvent values) for A into the equality A  1F = 1F  A = A

or

AF 1F = A and

1F FA = A

and multiplying the result by A−1 on suitable sides, we obtain the equalities F 1F = 1 and 1F F = 1, which mean that 1F = F −1 . A direct verification shows that the functions  · F and  · F are seminorms on B(F,G) (Y, X∗ ) equivalent to the initial norm on B(Y, X∗ ). The equality 1F F = F −1 F = 1 is obvious. Let us show that these seminorms are algebraic. The algebraicity of the seminorm  · F is a special case of [17, Theorem 10.2]. Clearly, for the seminorm  · F , we have AF ≤ AF . Therefore, A  BF = sup{γ(A  B) + (A  B)  C : |γ| + C ≤ 1} = sup{A  (γB + B  C) : |γ| + C ≤ 1} ≤ AF · sup{γB + B  C : |γ| + C ≤ 1} ≤ AF · BF ≤ AF · BF . Proposition 9. The resolvent of a pencil is a maximal F -pseudoresolvent; i.e., it cannot be extended over a set larger than ρ(F, G) to a pseudoresolvent satisfying Hilbert’s F -identity (2). MATHEMATICAL NOTES

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Proof. Suppose that, on the contrary, the resolvent of a pencil can be extended to a point λ ∈ / ρ(F, G) so that the extension satisfies Hilbert’s F -identity. Take any auxiliary point μ ∈ ρ(F, G). For λ and μ, Hilbert’s F -identity holds. Applying the transformations Rλ − Rμ = −(λ − μ)Rλ F Rμ , Rλ + (λ − μ)Rλ F Rμ = Rμ , Rλ ((μF − G)Rμ + (λ − μ)F Rμ ) = Rμ , Rλ (1X∗ + (λ − μ)F Rμ ) = Rμ ,  Rλ ((μF − G) + (λ − μ)F Rμ = Rμ , Rλ ((μF − G) + (λ − μ)F ) = 1X∗ , Rλ (λF − G) = 1X∗ , we see that Rλ is a left inverse of (λF − G). Similarly, (λF − G)Rλ = 1Y , i.e., Rλ is a right inverse of (λF − G). Thus, the point λ belongs to the resolvent set. The extended resolvent set of a pencil λ → λF − G is defined as a subset ρ(F, G) of the extended complex plane C consisting of ρ(F, G) and, possibly, ∞ [6, p. 31]. The point λ = ∞ is included in ρ(F, G) if the resolvent λ → (λF − G)−1 is defined in a punctured neighborhood of λ = ∞, the operator F is invertible, and limλ→∞ λRλ = F −1 in the norm of B(Y, X∗ ). Otherwise, the point λ = ∞ is included in the extended spectrum σ(F, G). Proposition 10. The point ∞ belongs to ρ(F, G) if and only if the operator F is invertible. By Theorem 8, in the most interesting case in which F has no inverse, the algebra B(F,G) (Y, X∗ ) has  (F,G) (Y, X∗ ) denote the algebra B(F,G) (Y, X∗ ) with adjoint unit I. In the case no unit. In this case, let B  (F,G) (Y, X∗ ) we mean the algebra B(F,G) (Y, X∗ ) itself and by I, the operator F −1 . of invertible F , by B Let K ⊆ C be a closed subset of the extended complex plane C, and let O(K) denote the set of all analytic functions f : U → C defined on open neighborhoods U of the set K (for different functions f , the neighborhoods U may be different). We say that two functions f1 : U1 → C and f2 : U2 → C are equivalent if the set K has an open neighborhood U ⊆ U1 ∩ U2 on which f1 and f2 coincide (that is, f1 (λ) = f2 (λ) for all λ ∈ U ). It is easy to show that this is indeed an equivalence relation. Thus, strictly speaking, the elements of O(K) are classes of equivalent functions. Obviously, O(K) is an algebra (without a norm) with unit u(λ) = 1.  (F,G) (Y, X∗ ) defined by Theorem 11. The map ϕ : O(σ(F, G)) → B ⎧  1 ⎪ ⎪ f (λ)(λF − G)−1 dλ if Γ does not enclose ∞, ⎨ 2πi Γ ϕ(f ) = 1 ⎪ ⎪ f (λ)(λF − G)−1 dλ + f (∞)I if Γ encloses ∞, ⎩ 2πi Γ where the contour Γ is the oriented envelope [18, p. 183 (Russian transl.)] of the extended spectrum σ(F, G) with respect to the complement of the domain of f , is a unital algebra morphism; in particular, ϕ(f g) = ϕ(f )  ϕ(g), and the function u(λ) = 1 is taken by ϕ to the unit I of the  (F,G) (Y, X∗ ). The morphism ϕ takes the function algebra B rλ0 (λ) =

1 , λ0 − λ

where

λ0 ∈ ρ(F, G),

to Rλ0 = (λ0 F − G)−1 , and if F is invertible, then ϕ takes v(λ) = λ to the operator F −1 GF −1 . For any function f ∈ O(σ(F, G)), σB 

(F,G) (Y,X∗ )

(ϕ(f )) = {f (λ) : λ ∈ σ(F, G)}.

Proof. The theorem is proved by standard arguments used in functional calculus [16, Chap. 1, Sec. 4, Theorem 3], [17, Theorem 10.27], [18, Theorems 5.2.5 and 5.11.2] and in spectral mapping theorems [18, Theorem 5.12.1]. MATHEMATICAL NOTES

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Theorem 12. If the set σ(F, G) decomposes into the disjoint union of two closed subsets σ0 , σ 1 ⊆ C and σ0 is bounded, then there exists an idempotent Π0 ∈ B(F,G) (Y, X∗ ) for which A = Π0  A  Π0 + Π1  A  Π1 ,

 (F,G) (Y, X∗ ), A∈B

where

Π1 = I − Π0 ,

the extended singular set of the pseudoresolvent Π0  R( · )  Π0 in the algebra Π0  B(F,G) (Y, X∗ )  Π0 coincides with σ0 , and the (extended) singular set of the pseudoresolvent Π1  R(·)  Π1 in the algebra Π1  B(F,G) (Y, X∗ )  Π1 coincides with σ1 = σ 1 \ {∞} (with σ 1 ). Proof. The proof of the theorem repeats that of spectral decomposition theorems (see, e.g., [16, Chap. 1, Sec. 4, Subsec. 4], [18, Theorem 5.13.1]. Proposition 13. The map A → AF is a morphism from the algebra B(F,G) (Y, X∗ ) to the algebra B(X∗ ), and the map A → FA is a morphism from the algebra B(F,G) (Y, X∗ ) to the algebra B(Y ). Corollary 14 stated below is the best known spectral fact in pencil theory (see, e.g., [3], [5], [6, Theorem 6.3], [8], [9], [11, Theorem 1.1.1], [14, Sec. 4.1], [15]). It readily follows from the properties  (F,G) (Y, X∗ ) described above and Proposition 13. of the algebra B Corollary 14. Under the assumptions of Theorem 12, the pairs of operators P0 = Π0 F ,

P1 = 1 − P0

and

Q0 = F Π0 ,

Q1 = 1 − Q0

are mutually complementary projectors and, therefore, they generate the direct sum decompositions X∗ = Im P0 ⊕ Im P1 ,

Y = Im Q0 ⊕ Im Q1 .

 (F,G) (Y, X∗ ), the relations P0 A = AQ0 and P1 A = AQ1 hold, which imply For any A ∈ B FP0 = Q0 F, GP0 = Q0 G,

FP1 = P1 F, GP1 = P1 G.

This means that the operators F and G have diagonal matrices with respect to the above direct sum decompositions. The extended spectrum of the pencil λ → λF − G : Im P0 → Im Q0 is equal to σ0 , and the extended spectrum of the pencil λ → λF − G : Im P1 → Im Q1 is equal to σ 1 . ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (grant no. 07-01-00131). REFERENCES 1. M. G. Krein and G. K. Langer, “Certain mathematical principles of the linear theory of damped vibrations of continua,” in Application of Theory of Functions in Continuum Mechanics, Proceedings of International Symposium, Tbilisi, 1963, Vol. 2: Fluid and Gas Mechanics, Mathematical Methods (Nauka, Moscow, 1965), pp. 283–322 [in Russian]. 2. N. I. Radbel’ and A. G. Rutkas, “Linear Operator Pencils and Noncanonical Systems,” Teor. Funktsii, Funktsional. Anal. i Prilozhen. 17, 3–14 (1973). 3. A. G. Rutkas, “The Cauchy problem for the equation Ax (t) + Bx(t) = f (t),” Differ.’nye Uravn. 11 (11), 1996–2010 (1975). 4. M. Povoas, “On some singular hyperbolic evolution equations,” J. Math. Pures Appl. (9) 60 (2), 133–192 (1981). MATHEMATICAL NOTES

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5. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, in Monogr. Textbooks Pure Appl. Math. (Marcel Dekker, New York, 1999), Vol. 215. 6. A. G. Baskakov and K. I. Chernyshov, “Spectral analysis of linear relations, and degenerate semigroups of operators,” Mat. Sb. 193 (11), 3–42 (2002) [Sb. Math. 193 (11), 1573–1610 (2002)]. 7. F. R. Gantmacher (Gantmakher), The Theory of Matrices (English translation of 1st Russian ed.: Chelsea, New York, 1959; Russian original, 4th ed.: Fizmatgiz, Moscow, 1967). 8. V. V. Ditkin, “Some spectral properties of a bundle of linear operators in Banach space,” Mat. Zametki 22 (6), 847–857 (1977) [Math. Notes, 22 (5–6), 965–971 (1977)]. 9. V. V. Ditkin, “Certain spectral properties of a pencil of linear bounded operators,” Mat. Zametki 31 (1), 75–79 (1982) [Math. Notes 31 (1–2), 39–41 (1982)]. 10. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils (Shtiintsa, Kishinev, 1986) [in Russian]. 11. G. A. Sviridyuk, “On the general theory of operator semigroups,” Uspekhi Mat. Nauk 49 (4), 47–74 (1994) [Russian Math. Surveys 49 (4), 45–74 (1994)]. 12. V. E. Fedorov, “Degenerate strongly continuous semigroups of operators,” Algebra Anal. 12 (3), 173–200 (2001) [St. Petersbg. Math. J. 12 (3), 471–489 (2001)]. 13. R. Cross, Multivalued Linear Operators, in Monogr. Textbooks Pure Appl. Math. (Marcel Dekker, New York, 1998), Vol. 213. 14. I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators.I, in Oper. Theory Adv. Appl. ¨ (Birkhauser, Basel, 1990), Vol. 49. 15. F. Stummel, “Diskrete Konvergenz linearer Operatoren. II,” Math. Z. 120, 231–264 (1971). ´ ements 16. N. Bourbaki, Theories ´ spectrales, in El ´ de mathematique ´ (Hermann, Paris, 1967; Mir, Moscow, 1972). 17. W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973; Mir, Moscow, 1975). 18. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, in Colloquium Publications (Amer. Math. Soc., Providence, RI, 1957; Inostr. Lit., Moscow, 1962).

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