Operators with numerical range in a conic domain

Beckermann, Bernhard; Crouzeix, Michel

doi:10.1007/s00013-007-1862-7

Cited by 6 publications

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References 8 publications

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“…We refer the reader also to the series of papers [3,5,[11][12][13]43], where the relationships between the notions of dilation, boundedness of functional calculus and numerical range are studied for general convex domains.…”

Section: Proof Of Theorem 51 (I) ⇔ (I )mentioning

confidence: 99%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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Abstract. The paper provides new characterisations of generators of cosine functions and C 0 -groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0 -groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini's result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh's characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0 -semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.

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Section: Proof Of Theorem 51 (I) ⇔ (I )mentioning

confidence: 99%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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“…Also, our Corollary 2.3 (b) may be improved for a sector. We refer the reader to [4] and [6,Chapter 4] for a discussion of optimized upper bounds as a function of θ. The strip case.…”

Section: The General Case Of Two Closed Disks Of the Riemann Spherementioning

confidence: 99%

“…From Theorem 1.1 we may conclude that the intersection of two disks of the Riemann sphere is a K-spectral set, with K ≤ 2 + 2/ √ 3. This includes the special case of a sector/strip obtained by the intersection of two half-planes and discussed in [7], and the lens-shaped intersection of two disks considered already in [5]; see also [4,6]. The case of the annulus is new and it permits to answer a question of A. Shields [20,Question 7] as described in the next paragraph.…”

Section: Theorem 11 Let a ∈ L(h) And Consider The Intersectionmentioning

confidence: 99%

“…Here p stands for "Poisson" and r for "residual". This decomposition reduces to the one used before in [3,7,6] in the special case ω = ∞ of a half-plane. The proof will consist in showing that the complete bounded norm of the map f → g p (f ) is bounded by n, while the complete bounded norm of the map f → g r (f ) can be estimated by n(n−1)/ √ 3.…”

Section: Decomposition Of the Cauchy Kernel Consider A Diskmentioning

confidence: 99%

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Intersections of several disks of the Riemann sphere as $K$-spectral sets

Badea

Beckermann

Crouzeix

2009

Communications on Pure &Amp; Applied Analysis

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We prove that if n closed disks D 1 , D 2 ,. . . , Dn, of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then their intersection D 1 ∩ D 2 · · · ∩ Dn is a complete K-spectral set for A, with K ≤ n + n(n − 1)/ √ 3. When n = 2 and the intersection X 1 ∩ X 2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974). 1 2 CATALIN BADEA, BERNHARD BECKERMANN, AND MICHEL CROUZEIX means that ρ A cb ≤ K. A complete spectral set is a complete K-spectral set with K = 1. Complete K-spectral sets are important in several problems of Operator Theory (see [16]).Spectral sets were introduced and studied by J. von Neumann [14] in 1951. In the same paper von Neumann proved that a closed disk {z ∈ C : |z − α| ≤ r} is a spectral set for A if and only if A − αI ≤ r. Also, the closed set {z ∈ C : |z − α| ≥ r} is spectral for A ∈ L(H) if and only if (A − αI) −1 ≤ r −1 , and the half-plane {Re(z) ≥ 0} is a spectral set if and only if Re Av, v ≥ 0 for all v ∈ H. Therefore for any closed disk D of the Riemann sphere (interior/exterior of a disk or a half-plane) it is easy to check whether D is a spectral set.We refer to [19] for a treatment of the von Neumann's theory of spectral sets and to [15] and the book [16] for modern surveys of known properties of K-spectral and complete K-spectral sets.The problem. Let X be the intersection of n disks of the Riemann sphere, each of them being a spectral set for a given operator A ∈ L(H). In the present paper we will be concerned with the question whether X itself is a (complete) K-spectral set for A.The intersection of two spectral sets is not necessarily a spectral set; counterexamples for the annulus are presented in [24,13,15]. However, the same question for K-spectral sets remains open. The counterexamples for spectral sets show that the same constant cannot be used for the intersection.Some cases of the K-spectral set problem have been solved. If two K-spectral sets have disjoint boundaries, then by a result of Douglas and Paulsen [9] the intersection is a K ′ -spectral set for some K ′ . The case when the boundaries meet was considered by Stampfli [21,22] and Lewis [12]. In particular, it is proved in [12] that the intersection of a (complete) K-spectral set for the bounded linear operator A with the closure of any open set containing the spectrum of A is a (complete) K ′ -spectral set for A.The main result. In this paper we will show the following result, which gives a positive answer to a question raised by Michael A. Dritschel (personal communication). Theorem 1.1. Let A ∈ L(H), and consider the intersection X = D 1 ∩ D 2 ∩ · · · ∩ D n of n disks of the Riemann sphere C, each of them being spectral for A. Then X is a complete K-spectral set for A, with a constant K ≤ n + n(n−1)/ √ 3.

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“…The second bound is better than the first if α ≤ .22 π and is still valid if we replace the sector S α by (a domain limited by) a branch of hyperbola of angle 2α. In [4] we derived the bound C cb (E) ≤ 2 + 2/ √ 4−e 2 for an ellipse E of eccentricity e and C cb (P) ≤ 2+2/ √ 3 for a parabola P. The estimate C cb (S 0 ) ≤ 2+2/ √ 3 is also known for a strip S 0 , [13].…”

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confidence: 98%

Some Constants Related to Numerical Ranges

Crouzeix¹

2016

SIAM J. Matrix Anal. & Appl.

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In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2-spectral set for the matrix A, we propose a study of various constants. We review some partial results; many problems are still open. We describe our corresponding numerical tests.• for all square matrices A ∈ C d,d , for all values of d, • for all polynomial functions P : C → C m,n , for all values of m and n.Here P is matrix-valued P (z) = (p ij (z)), with each entry p ij ∈ C[z] being a polynomial ; the matrix P (A) ∈ C md,nd is constituted of m × n blocks of size d × d, the (i, j)-th block being p ij (A).The surprising fact is the existence of such uniform bounds Q and Q cb , independently of the matrix A, of its size, of the degree of polynomials used, as well as of m

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Operators with numerical range in a conic domain

Cited by 6 publications

References 8 publications

Resolvent characterisation of generators of cosine functions and C 0-groups

Resolvent characterisation of generators of cosine functions and C 0-groups

Intersections of several disks of the Riemann sphere as $K$-spectral sets

Some Constants Related to Numerical Ranges

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