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Banach J. Math. Anal. 6 (2012), No. 1, 45–60 LINEAR MAPS

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Banach J. Math. Anal. 6 (2012), No. 1, 45–60 LINEAR MAPS Banach J. Math. Anal. 6 (2012), no. 1, 45–60 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/BJMA/ LINEAR MAPS PRESERVING PSEUDOSPECTRUM AND CONDITION SPECTRUM G. KRISHNA KUMAR1 AND S. H. KULKARNI2 Communicated by K. Jarosz Abstract. We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Sev- eral results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A, B be complex unital Banach algebras and > 0. Let Φ : A → B be an -pseudospectrum preserving linear onto map. Then Φ preserves spectrum. If A and B are uniform algebras, then, Φ is an isometric isomorphism. (2) Let A, B be uniform algebras and 0 < < 1. Let Φ : A → B be 0 an -condition spectrum preserving linear map. Then Φ is an -almost 0 multiplicative map, where , tend to zero simultaneously. 1. Introduction Let A be a complex Banach algebra with unit 1. We shall identify λ.1 with λ. We recall that the spectrum of an element a ∈ A is defined as −1 σ(a) = λ ∈ C : λ − a∈ / A , where A−1 is the set of all invertible elements of A. The spectral radius of an element a is defined as r(a) = sup{|λ| : λ ∈ σ(a)}. Date: Received: 20June 2011; Revised: 25 August 2011; Accepted: 8 September 2011. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 47B49; Secondary 46H05, 46J05, 47S48. Key words and phrases. Pseudospectrum, condition spectrum, almost multiplicative map, linear preserver, perturbation. 45 46 G. KRISHNA KUMAR, S.H. KULKARNI There are several extensions and generalizations of the concept of spectrum. These include the Ransford’s generalized spectrum [18], -pseudospectrum [21] and the -condition spectrum [13]. Unlike the spectrum, which is a purely alge- braic concept, both the -pseudospectrum and -condition spectrum depend on the norm. Also both these sets contain the spectrum as a subset. Linear preserver problems (LPP) is an active research area in matrix and operator theory. A brief discussion on LPP can be found in [3]. The monograph by Molnar [15] contains a wealth of information about such problems. The most popular among these is the problem of characterizing spectrum preserving maps, studied by many authors [14, 7, 20]. Let X, Y be Banach spaces, BL(X) denote the algebra of all bounded linear operators on X and X∗ denote the dual of X. Jafarian and Sourour proved [7] that a spectrum preserving linear map Φ from BL(X) onto BL(Y ) is either of the form Φ(T ) = AT A−1,T ∈ BL(X) for an isomorphism A from X onto Y or of the form Φ(T ) = BT ∗B−1,T ∈ BL(X) for an isomorphism B of X∗ onto Y . Thus Φ is multiplicative or anti-multiplicative. In this paper we study the linear maps which preserve -pseudospectrum and -condition spectrum between complex unital Banach algebras. One of the sur- prises here is that it turns out that a map preserving -pseudospectrum also preserves spectrum (Theorem 3.10). In many cases such a map simply becomes an isometric isomorphism (Remark 3.12). In section 2, we give definitions of -pseudospectrum, -condition spectrum, - almost multiplicative map and -isometry. We establish relations between pseu- dospectrum and condition spectrum of an element in a complex unital Banach algebra (Proposition 2.5, 2.6). In Section 3, we prove that any linear map between complex unital Banach algebras which preserves -pseudospectrum also preserves spectrum (Theorem 3.10). We also prove an analogue of the Gleason–Kahane– Zelazko theorem for -pseudospectrum (Theorem 3.13). In section 4, we discuss -condition spectrum preserving maps in complex unital Banach algebras. We establish that -condition spectrum is closely related to -almost multiplicative map (Theorem 4.4). In section 5, we study -perturbation of a complex unital Banach algebra. We prove various properties and relations between spectrum, pseudospectrum and condition spectrum of an element in a Banach algebra and its -perturbation. 2. Preliminaries In this section we introduce some definitions and terminology used in this paper. A relation connecting pseudospectrum and condition spectrum of an element in a complex unital Banach algebra is given. We also show that a small perturbation of an isomorphism is an almost multiplicative map. Definition 2.1. (-pseudospectrum) Let A be a complex unital Banach algebra with unit 1 and > 0. The -pseudospectrum of an element a ∈ A is denoted by Λ(a) and is defined as, 1 Λ (a) := λ ∈ : k(λ − a)−1k ≥ , C PSEUDOSPECTRUM AND CONDITION SPECTRUM 47 with the convention that k(λ − a)−1k = ∞ if λ − a is not invertible. Note that because of this convention, σ(a) ⊆ Λ(a) for every > 0. For more information on -pseudospectrum one may refer to [21]. Definition 2.2. (-condition spectrum) Let A be a complex unital Banach alge- bra with unit 1 and 0 < < 1. The -condition spectrum of an element a ∈ A is denoted by σ(a) and is defined as, 1 σ (a) := λ ∈ : kλ − akk(λ − a)−1k ≥ , C with the convention that kλ − akk(λ − a)−1k = ∞, if λ − a is not invertible. Here also σ(a) ⊆ σ(a) for 0 < < 1. One may refer to [13] for examples and elementary properties of the -condition spectrum. Definition 2.3. Let A, be as in Definition 2.2. The -condition spectral radius r(a) of an element a ∈ A is defined as r(a) := sup{|λ| : λ ∈ σ(a)}. Remark 2.4. Let A, be as in Definition 2.2 then, 1 + r(a) ≤ r (a) ≤ kak. 1 − (see Theorem 2.9 of [13]). Next two propositions establish a relationship between condition spectrum and pseudospectrum of an element in a complex unital Banach algebra. These propo- sitions provide an answer to a question raised by the authors in [1] (see Remark 4.4 of [1]). Proposition 2.5. Let A be a complex Banach algebra with unit 1, a ∈ A and 0 < < 1. Then σ(a) ⊆ Λ 2kak (a) . 1− (1 + )kak Proof. Let λ ∈ σ (a). Then |λ| ≤ and hence 1 − (1 + )kak 2kak kλ − ak ≤ |λ| + kak ≤ + kak = . 1 − 1 − Since λ ∈ σ(a), we have 1 kλ − akk(λ − a)−1k ≥ . Thus 1 k(λ − a)−1k ≥ kλ − ak 1 − ≥ . 2kak 48 G. KRISHNA KUMAR, S.H. KULKARNI Proposition 2.6. Let A be a complex Banach algebra with unit 1 and > 0. Suppose a ∈ A is not a scalar multiple of 1 and let Ma := inf{kz − ak : z ∈ }. Then Λ(a) ⊆ σ (a). C Ma Proof. Suppose λ ∈ Λ(a). Then, 1 k(λ − a)−1k ≥ . Also, kλ − ak ≥ inf{kz.1 − ak : z ∈ C} = Ma > 0. Hence M kλ − akk(λ − a)−1k ≥ a . Remark 2.7. If a = µ.1 for some µ ∈ C, then -condition spectrum of a is the singleton set {µ} and -pseudospectrum is the closed ball with center µ and radius . Thus the condition on a can not be dropped from the above proposition. Definition 2.8. Let A, B be complex Banach algebras and > 0. A linear map Φ: A → B is said to be (1) an -almost multiplicative map, if kΦ(ab) − Φ(a)Φ(b)k ≤ kakkbk for all a, b ∈ A. (2) an -almost anti-multiplicative map, if kΦ(ab) − Φ(b)Φ(a)k ≤ kakkbk for all a, b ∈ A. (3) an -almost Jordan multiplicative map, if kΦ(a2) − Φ(a)2k ≤ kak2 for all a ∈ A. See [8] for more information on such maps. In [8] -almost multiplicative maps are called -isomorphisms. We have avoided this terminology because the word isomorphism usually includes the assumption of injectivity. We have not made such an assumption here. If B = C, Φ is called -almost multiplicative linear functional. It is obvious that every -almost multiplicative or - almost anti- multiplicative map is - almost Jordan multiplicative map. See [1] for a detailed discussion of the converse of this, called approximate Herstein’s theorem. Definition 2.9. (-isometry) Let A, B be complex Banach spaces and > 0. A linear continuous injective map Φ : A → B is said to be an -isometry , if kΦk ≤ 1 + and kΦ−1k ≤ 1 + . One may refer to [8] for more information on - isometry. Example 2.10. (-almost multiplicative map) Consider C2×2 as the set of all 1 bounded linear maps on ( 2, k · k ). For 0 < < consider Φ : 2×2 → 2×2 C 2 4 C C defined by a a a a (1 + ) Φ( 11 12 ) = 11 12 . a21 a22 a21(1 + ) a22 PSEUDOSPECTRUM AND CONDITION SPECTRUM 49 We can write the above as Φ(A) = A + Ψ(A), where Ψ : C2×2 → C2×2 is a linear map given by a a 0 a Ψ( 11 12 ) = 12 . a21 a22 a21 0 So that kΨk ≤ . Now for A, B ∈ C2×2 we have kΦ(A.B) − Φ(A).Φ(B)k = kΨ(A.B) − Ψ(A).B − A.Ψ(B) − Ψ(A).Ψ(B)k ≤ 4kΨkkAkkBk ≤ 4kAkkBk. Thus Φ is a 4-almost multiplicative map. The following proposition shows that a small perturbation of a homomorphism is an almost multiplicative map. Proposition 2.11. Let A, B be complex Banach algebras, Φ: A → B be a continuous homomorphism, 0 < < 1 and Ψ: A → B a bounded linear map with kΨk ≤ min , . Then Φ + Ψ is an -almost multiplicative map. 4 4kΦk Proof. For a, b ∈ A we have, k(Φ + Ψ)(ab) − (Φ + Ψ)(a)(Φ + Ψ)(b)k = kΦ(ab) + Ψ(ab) − (Φ(a) + Ψ(a))(Φ(b) + Ψ(b))k = kΦ(ab) − Φ(a)Φ(b) + Ψ(ab) − Ψ(a)Ψ(b) − Φ(a)Ψ(b) − Ψ(a)Φ(b)k ≤ (kΨk + kΨk2 + 2kΦkkΨk)kakkbk ≤ kakkbk.
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