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Lemniscates and K-Spectral Sets View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Journal of Functional Analysis 262 (2012) 1728–1741 www.elsevier.com/locate/jfa Lemniscates and K-spectral sets Olavi Nevanlinna Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 11100, Otakaari 1M, Espoo, FI-00076 Aalto, Finland Received 30 May 2011; accepted 15 November 2011 Available online 25 November 2011 Communicated by S. Vaes Abstract We show how multicentric representation of functions provides a simple way to generalize the von Neu- mann result that the unit disc is a spectral set for contractions in Hilbert spaces. In particular the sets need not be connected and the results can be applied to bounding Riesz spectral projections. © 2011 Elsevier Inc. All rights reserved. Keywords: von Neumann spectral sets; K-spectral sets; Lemniscates; Multicentric representation; Jacobi series; Riesz spectral projections 1. Multicentric representation of holomorphic functions 1.1. Motivation If A denotes a bounded operator in a Hilbert space, we denote Vp(A) = z ∈ C: p(z) p(A) (1.1) where p is a monic polynomial with distinct roots. We shall show that these sets are K-spectral, whenever the lemniscate does not pass through any critical point of p. As any compact set can be the spectrum of a bounded operator, it is crucial that lemniscates do have good approxima- tion properties, a topic which was started by D. Hilbert in 1897, see e.g. [11]. Furthermore, we have recently provided an algorithm [5] which, for any given bounded A, produces a sequence E-mail address: Olavi.Nevanlinna@aalto.fi. 0022-1236/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2011.11.019 O. Nevanlinna / Journal of Functional Analysis 262 (2012) 1728–1741 1729 of monic polynomials p with distinct roots such that the sets Vp(A) squeeze around the polyno- mially convex hull of the spectrum of A. In order to be able to estimate a holomorphic function f effectively at an operator A we use the approach we introduced in [6], which one may view as a combination of Jacobi series [4,11] with Lagrange interpolation. Each monic polynomial d p(z) = (z − λj ) j=1 with simple roots λj induces a unique multicentric representation of f , d f(z)= δk(z)fk(w), with w = p(z). (1.2) k=1 Here δk denote the polynomials of degree d − 1 taking the value 1 at λk and vanishing at the other roots. In [6] we have also discussed the practical computation of the Taylor series of fk. In fact, the coefficients can be computed in a recursive fashion if the derivatives of the original function f are available at the local centers λk. 1.2. Key estimate The representation (1.2) allows an obvious avenue for analysis, estimation and computation in complicated sets. One just treats the functions fk in discs |w| R and combines the estimates for f in the sets satisfying |p(z)| R. In this paper we demonstrate this approach by generalizing a well-known result of von Neu- mann on contractions in Hilbert spaces. In order to do this we need to have an estimate of the following form sup fk(w) C(R) sup f(z) . (1.3) |w|R |p(z)|R Such an estimate would then imply that the sets Vp(A) are K-spectral sets with some K. In order to state it we need some notation. Let γR denote the lemniscate γR = z ∈ C: p(z) = R . For small R the lemniscate consists of d separate circular curves, for large R it reduces to just one circular curve. In general the lemniscate is smooth except if it contains a critical point, where the derivative of p vanishes. Thus there are at most d − 1 such exceptional values R.Lets(R) denote the distance from γR to the set of critical points. Theorem 1.1. If p is a monic polynomial of degree d with distinct roots, then there exists a constant C such that if f is holomorphic for |p(z)| R, then the functions fk in (1.2) are 1730 O. Nevanlinna / Journal of Functional Analysis 262 (2012) 1728–1741 holomorphic for |w| R and if γR does not contain any critical points of p the estimate (1.3) holds with some C(R) satisfying C C(R) 1 + . (1.4) s(R)d−1 Remark 1.2. If C(R) denotes the smallest constant such that (1.3) holds for all f then C(R) → 1 as R → 0orR →∞. Generically the critical points are simple and then the constant is propor- tional to 1/s(R) but we include an example where the behavior is of the form 1/s(R)d−1. We postpone the proof while first giving applications to the spectral set theory. 2. Applications to K-spectral sets 2.1. K-spectral sets using the von Neumann theorem We recall the definitions related to this topic. We denote by B(H) the space of bounded linear operators in a Hilbert space H . Definition 2.1. AclosedsetΣ ⊂ C is a spectral set for A ∈ B(H), if for all rational functions r with poles off Σ there holds r(A) supr(z). (2.1) z∈Σ If the equation holds in the form r(A) K supr(z), z∈Σ withafixedK, then Σ is called a K-spectral set. The topic began with a fundamental result by von Neumann for contractions in Hilbert spaces. Theorem 2.2. (See von Neumann (1951) [10].) If A ∈ B(H), and A 1, then the closed unit disc is a spectral set for A. This can clearly be reformulated also as follows: f(A) sup f(z) (2.2) |z|A provided f is holomorphic in |z| A. We formulate our results for holomorphic functions rather than for polynomials or rational functions as we consider sets which may consist of several simply connected components. In particular, then the results apply as such for Riesz spectral projections. Here is the main result of this paper. O. Nevanlinna / Journal of Functional Analysis 262 (2012) 1728–1741 1731 Theorem 2.3. Suppose we are given a monic p ∈ Pd with distinct roots and a bounded operator A ∈ B(H) in a Hilbert space H . Let R 0 satisfy p(A) R and be such that the lemniscate γR contains no critical points of p. Then for all f which are holomorphic for |p(z)| R there holds f(A) K sup f(z), (2.3) |p(z)|R where the constant K satisfies d K C(R) δk(A) , (2.4) k=1 with C(R) as in Theorem 1.1. Proof. The claim follows immediately from Theorem 1.1 and from the von Neumann Theo- rem 2.2. In fact, denoting B = p(A) we have from (2.2) fk(B) sup fk(w) |w|R and so by Theorem 1.1 fk(B) C(R) sup f(z) . |p(z)|R Then the result follows from d f(A)= δk(A)fk(B). 2 k=1 2.2. Application to the Riesz spectral projections A simple but useful application of the previous result is obtained as follows. Suppose γR consists of several components and is free from critical points. Then one can define f to be iden- tically 1 in some open neighborhood of some of the components and to vanish in a neighborhood of all the others. If A ∈ B(H) is such that p(A) R, then the resulting operator is simply the Riesz spectral projection to the invariant subspace w.r.t. the part of the spectrum where f equals 1. The following example shows that the constant C(R) of Theorem 1.1 has to blow up near the critical lemniscates, and that the worst behavior in (1.4) may happen. Example 2.4. Let ε>0 be small. Consider the matrix ε 1 A(ε) = , (2.5) 0 −ε 1732 O. Nevanlinna / Journal of Functional Analysis 262 (2012) 1728–1741 with spectrum σ(A(ε))={ε, −ε}.Letp(λ) = λ2 − 1 so that we have one critical point at the origin. Put R = 1 − ε2 so that the spectrum lies on the boundary of the lemniscate |p(z)|= p(A(ε))=1 − ε2.Letf be 1 on the right open half plane and 0 on the left open half plane, so in particular it is holomorphic inside and in a neighborhood of the lemniscate. Then f(A(ε))= P(ε)is well defined and equals the Riesz spectral projection onto the direction of the eigenvector w.r.t. the eigenvalue ε. In fact, the resolvent satisfies −1 1 λ + ε 1 λI − A(ε) = . λ2 − ε2 0 λ − ε The Riesz projector can be obtained as the residue at ε: 11/2ε P(ε)= . 00 As the distance from γR to the critical point is ε,wehave 1/2 P(ε) ∼ . s(R) Likewise, if p(λ) = λd − 1 we could take R = 1 − εd and e.g. the truncated backward shift and perturb it slightly: ⎛ ⎞ 01 ⎜ ⎟ ⎜ 01 ⎟ ⎜ ⎟ = ⎜ . ....⎟ S(ε) ⎜ ⎟ . ⎝ . ....⎠ 01 εd 0 ..0 Again, the eigenvalues are at distance ε from the origin and the projection to the direction of an eigenvector behaves like 1/d P(ε) ∼ . s(R)d−1 In fact, ⎛ ⎞ εd−1 εd−2 .. 1 ⎜ εd εd−1 .. ε⎟ 1 ⎜ ⎟ P(ε)= ⎜ .....⎟ . d−1 ⎝ ⎠ dε ..... ε2d−2 ε2d−3 ..εd−1 In this case we would take the analytic function f to be identically 1 in the component which contains the point 1 and in the others we set it equal to zero.
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