PSEUDOSPECTRA OF SPECIAL OPERATORS AND PSEUDOSECTRUM PRESERVERS JIANLIAN CUI, CHI-KWONG LI, AND YIU-TUNG POON Abstract. Denote by B(H) the Banach algebra of all bounded linear operators on a complex Hilbert space H. Let A 2 B(H), and denote by σ(A) the spectrum of A. For " > 0, define the "-pseudospectrum σ"(A) of A as σ"(A) = fz 2 σ(A + E): E 2 B(H); kEk < "g: In this paper, the pseudospectra of several special classes of operators are characterized. As an application, complete descriptions are given of the maps of B(H) leaving invariant the pseudospectra of A • B for different kind of binary operations • on operators such as the difference A − B, the operator product AB, and the Jordan product AB + BA. 1. Introduction Denote by B(H) the Banach algebra of all bounded linear operators on a complex Hilbert space H. If H has dimension n < 1 we identify B(H) as the algebra Mn of n × n complex matrices. Let A 2 B(H) and σ(A) = fz 2 C : zI − A is not invertible in B(H)g the spectrum of A. The spectrum of an operator provides a lot of useful information about P m k it. For instance, limm!1 k=0 A exists if σ(A) lies inside a disk centered at the origin with d radius r < 1; a system of differential equations governed by dt x = Ax for a given matrix A always has an equilibrium solution if σ(A) ⊆ fz :(z + z∗) < 0g: So, there is a lot of interest in finding efficient ways to determine or estimate σ(A). Moreover, due to numerical and measuring errors, and also due to the fact that σ(A) is very sensitive to perturbation, researchers propose the study of the "-pseudospectrum of A for a given " > 0 defined by σ"(A) = [fσ(A + E): E 2 B(H); kEk < "g: (1:1) 2010 Mathematics Subject Classification. Primary 47B48, 46L10. Key words and phrases. operator; pseudospectrum; preserver problems. Research of the first author was supported by National Natural Science Foundation of China (No.11271217, 10871111). Research of the second and third authors was partially supported by USA NSF and HK RGC. 1 2 JIANLIAN CUI, CHI-KWONG LI, AND YIU-TUNG POON Here kEk is the (spectral) norm of E. Evidently, for " 2 (0; 1), the "-pseudospectra of A form a family of strictly nested closed sets, which grow to fill the whole complex plane as " ! 1. It follows from the upper-semicontinuity of the spectrum that the intersection of all the pseudospectra is the spectrum, \">0σ"(A) = σ(A): The operator (zI − A)−1 is the resolvent of an operator A at the point z 2 C. One may also define the "-pseudospectrum as follows: −1 −1 σ"(A) = fz 2 C : k(zI − A) k > " g: (1:2) Here we use the convention that k(zI − A)−1k = 1 if z 2 σ(A). Two other equivalent definitions of the "-pseudospectrum are respectively σ"(A) = fz 2 C : k(zI − A)xk < " for some unit vector x 2 Hg; (1:3) and σ"(A) = fz 2 C : smin(zI − A) < "g; (1:4) where smin denotes the minimal singular value in the matrix case or the smallest s-number for an operator [7]. Clearly, the pseudospectrum is invariant under unitary similarities. Denote by At the transpose of A relative to an arbitrary but fixed orthonormal basis of H. Then t t σ"(A) = σ"(A ) because σ(X) = σ(X ) for any X 2 B(H). Let us recall other properties of the pseudospectrum (see [12]), which will be frequently used in our proof. Let " > 0 be arbitrary and D(a; ") = fµ 2 C : jµ − aj < "g, where a 2 C. Property 1.1. Let " > 0 and let A 2 B(H). (1) σ(A) + D(0;") ⊆ σ"(A). (2) If A is normal, then σ"(A) = σ(A) + D(0;"). (3) For any c 2 C, σ"(A + cI) = c + σ"(A). (4) For any nonzero c 2 C, σ"(cA) = cσ " (A). jcj In this paper, we give complete descriptions of some special classes of operators in terms of the pseudospectrum in Section 2. For example, we prove that an operator is a multiple of a self-adjoint operator if and only if its pseudospectrum lies in the set fz 2 C : j=(z)j < "g. In particular, an operator A = αI for some scalar α if and only if σ"(A) = D(α; "); an operator PSEUDOSPECTRA AND PRESERVER PROBLEMS 3 A is a nontrivial projection if and only if σ"(A) = D(0;") [ D(1;"). Furthermore, we show p 2 2 that if an operator A satisfies A = 0, then σ"(A) = D(0; " + kAk"). In particular, if A p 2 is of rank one, then A is nilpotent if and only if σ"(A) = D(0; " + kAk"); we also prove that the map ("; A) 7! σ"(A) is continuous. In Sections 3{5, we characterize the maps Φ on operators such that σ"(A•B) = σ"(Φ(A)•Φ(B)), where A•B is one of the binary operations: A−B; AB; AB+BA. We note that the study of similar problems on matrix algebras was done in [4]. In this paper, we develop additional tools to lift the results to the infinite dimensional case, and treat the Jordan product AB + BA. Linear preservers of pseudospectrum have also been studied in a recent paper by Kumar and Kulkarni [8]. To conclude this section, let us fix some notations. Let H be a complex Hilbert space. ∗ Denote by B(H) the C -algebra of all bounded linear operators on H and by Bs(H) the set of all self-adjoint operators in B(H). For A 2 B(H), ker A and rngA denote the kernel and range of A, respectively. For a closed subspace M of H, AjM denotes a restriction of A to M, ? and M denotes the orthogonal complement of M in H, and PM the orthogonal projection from H onto M. For x 2 H,[x] denotes the linear space spanned by x. For any nonzero x; f 2 H, denote by x ⊗ f rank one operator z 7! hz; fix, and all rank one operators in B(H) can be written into this form. Denote by I the identity operator in B(H). Let TrA denote the trace of a finite rank operator A. 2. Some properties of the pseudospectrum In this section, we give complete descriptions of the pseudospectra of self-adjoint operators. We also classify the pseudospectra of several classes of operators such as projections and square zero operators. Moreover, we obtain a result concerning the continuity of the pseudospectrum. Lemma 2.1. Let " > 0 and A 2 B(H). Assume that α 2 σp(A) (the set of all point spectra of A). If ker(αI − A) is not a reduced subspace of A, then there exists r > " such that D(α; r) ⊂ σ"(A). Proof. Let α 2 σp(A), then ker(αI − A) is an invariant subspace of A. According to the space decomposition H = ker(αI−A)⊕ker(αI−A)?, A has an operator matrix representation 0 1 αI B A = @ A : 0 C 4 JIANLIAN CUI, CHI-KWONG LI, AND YIU-TUNG POON Since ker(αI − A) is not a reduced subspace of A, we have B =6 0. Therefore there exist orthogonal unit vectors u 2 ker(A − αI)? and v 2 ker(A − αI) such that hBu; vi 6= 0. Let ∗ P = u ⊗ u + v ⊗ v. For each z 2 D(0;"), take αz = α + z. Then P (αzI − A) (αzI − A)P is unitarily similar to Bz ⊕ 0, where 0 1 j j2 − h i @ z z Bu; v A Bz = : 2 2 −zhv; Bui jhBu; vij + k(αzI − C)uk ∗ 2 Thus the minimal eigenvalue λz of P (αzI − A) (αzI − A)P is smaller than jzj , and hence p there exists a unit vector xz 2 [u; v] such that k(αzI − A)P xzk = λz < jzj ≤ ". Since the map z ! λz is continuous on D(0;"), there exists d < " such that k(αzI − A)P xzk ≤ d for all 3" − d z 2 D(0;"). Let r = > ". For every z 2 D(0; r), consider the following cases: 2 Case 1 jzj ≤ ". From the above discussion, we have a unit vector xz 2 [u; v] such that k(αzI − A)P xzk ≤ d < ". Hence, α + z 2 σ"(A). it it Case 2 jzj > ". Let z = jzje with t 2 R. Let "t = "e . From the above discussion, we 2 k − k ≤ have a unit vector x"t [u; v] such that (α"t I A)P x"t d < ". We have k − k k it j j − it − k ((α + z)I A)x"t = (α + "e + ( z ")e )I A)x"t " + d ≤ k(α I − A)P x k + jzj − " ≤ d + r − " = < " "t "t 2 Hence, α + z 2 σ"(A). Thus, D(α; r) ⊂ σ"(A) and the proof is complete. Theorem 2.2. Let " > 0, A 2 B(H), and t 2 R. Then eitA is self-adjoint if and only if it σ"(A) ⊂ fz 2 C : jIm e zj < "g. it it Proof. Since σ"(e A) = e σ"(A), it suffices to prove the case when t = 0. Let A 2 B(H) be any self-adjoint operator. Since σ"(A) = [α2σ(A)D(α; "), it follows that for every z 2 σ"(A), jIm zj < ".