Spectral Unit Ball Research Articles

RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows — generalizing a result of Ransford–White for the spectral unit ball — that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider [Formula: see text]-point and [Formula: see text]-point Pick–Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

Lifting Maps from the Symmetrized Polydisc in Small Dimensions

The spectral unit ball $\Omega_n$ is the set of all $n\times n$ matrices with spectral radius less than $1$. Let $\pi(M) \in \mathbb C^n$ stand for the coefficients of its characteristic polynomial of $M$ (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisk is $\mathbb G_n:=\pi(\Omega_n)$. When investigating Nevanlinna-Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisk (for instance to obtain continuity results for the Lempert function): if $\psi \in \mathcal O(\mathbb D, \Omega_n)$, then $\pi \circ \psi \in \mathcal O(\mathbb D, \mathbb G_n)$. Given a map $\varphi \in \mathcal O(\mathbb D, \mathbb G_n)$, we are looking for necessary and sufficient conditions for this map to "lift through given matrices", i.e. find $\psi$ as above so that $\pi \circ \psi = \varphi$ and $\psi (\alpha_j) = M_j$, $1\le j \le N$. A natural necessary condition is $\varphi(\alpha_j)=\pi(M_j)$, $1\le j \le N$. When the matrices $M_j$ are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of $\varphi$ at the points $\alpha_j$. Those conditions are necessary and sufficient for a local lift. We give a scheme which shows that the necessary conditions are also sufficient for a global lift in small dimensions (up to $5$), and a counter-example to show that the scheme fails in dimension $6$ (and above).

A Family of Domains Associated with $${\mu}$$ μ -Synthesis

We introduce a family of domains --- which we call the $\mu_{1,n}$-quotients --- associated with an aspect of $\mu$-synthesis. We show that the natural association that the symmetrized polydisc has with the corresponding spectral unit ball is also exhibited by the $\mu_{1,n}$-quotient and its associated unit "$\mu_E$-ball". Here, $\mu_E$ is the structured singular value for the case $E = \{[w]\oplus(z I_{n-1})\in \mathbb{C}^{n\times n}: z,w\in \mathbb{C}\}$, n = 2, 3, 4,... Specifically: we show that, for such an $E$, the Nevanlinna-Pick interpolation problem with matricial data in a unit "$\mu_E$-ball", and in general position in a precise sense, is equivalent to a Nevanlinna-Pick interpolation problem for the associated $\mu_{1,n}$-quotient. Along the way, we present some characterizations for the $\mu_{1,n}$-quotients.

On the automorphisms of the spectral unit ball

On the automorphisms of the spectral unit ball

Separate continuity of the Lempert function of the spectral ball

We find all matrices A from the spectral unit ball Ω n such that the Lempert function l Ω n ( A , ⋅ ) is continuous.

Commuting holomorphic maps on the spectral unit ball

We prove that if F is a holomorphic map from the open spectral unit ball of a primitive Banach algebra into itself satisfying F (0) = 0, F ′ (0) = I and F (x) x = xF (x) for every x, then F is the identity map. Using this, we prove that if A is a semisimple Banach algebra and B is a primitive Banach algebra, then any unital spectral isometry from A onto B which locally preserves commutativity is a Jordan morphism. The same is true when A and B are both assumed to be von Neumann algebras.

A local form for the automorphisms of the spectral unit ball

If F is an automorphism of Ω n , then 2-dimensional spectral unit ball, we show that, in a neighborhood of any cyclic matrix of Ωn, the mapF can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.

Proper holomorphic mappings of the spectral unit ball

We prove an Alexander type theorem for the spectral unit ball Ω n showing that there are no non-trivial proper holomorphic mappings in Ω n , n > 2.

On the zero set of the Kobayashi–Royden pseudometric of the spectral unit ball

Given $A\in\Omega_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a generalized tangent vector at $A$ to an entire curve in $\Omega_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $\Omega_3,$ the zero set of this metric is completely described.

Some New Observations on Interpolation in the Spectral Unit Ball

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball \Omega_n, n\geq 2. We begin by showing that a known necessary condition for the existence of a $\mathcal{O}(D;\Omega_n)$-interpolant (D here being the unit disc in the complex plane), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem -- one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of \Omega_n, n\geq 2.

A Cartan type theorem for finite-dimensional algebras

Let A be a finite direct sum of full matrix algebras over the complex field. We prove that if F is a holomorphic map of the open spectral unit ball of A into itself such that F ( 0 ) = 0 and F ′ ( 0 ) = I , the identity of A , then a and F ( a ) have always the same spectrum. As an application we obtain a new proof, purely function-theoretic, of the fact that a unital spectral isometry on a finite-dimensional semi-simple Banach algebra is a Jordan morphism.

THE $2\times 2$ SPECTRAL NEVANLINNA–PICK PROBLEM

A method is presented to construct interpolation functions into the 2 × 2 open spectral unit ball. For the spectral Nevanlinna–Pick problem, these functions are in some sense extremal, and the set of all these interpolation functions is enough to solve any interpolation problem, with solvable finite interpolation data. This fact is used to compute the complex geodesics for the symmetrized bidisc and for the spectral unit ball, and to solve completely the two-point interpolation problem for the two target sets.

Geometry of the symmetrized polydisc

We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in \(\mathcal{M}_n (\mathbb{C}^n ).\)

On the automorphisms of the spectral unit ball

Let ${\mit \Omega }$ be the spectral unit ball of $M_n({\mathbb C})$, that is, the set of $n\times n$ matrices with spectral radius less than 1. We are interested in classifying the automorphisms of ${\mit \Omega }$. We know that it is enough to consider

Non-Linear Spectrum-Preserving Maps

Let Mn denote the space of complex n×n matrices, and let Ωn denote the spectral unit ball of Mn, namely the set of matrices in Mn whose eigenvalues lie within the open unit disc. As a step towards the eventual classification of the holomorphic automorphisms of Ωn, we prove that every such automorphism F satisfying F(0) = 0 and F′(0) = I has the property that F(x) is conjugate to x for each x∈Ωn. This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum-preserving maps proved in this paper. 1991 Mathematics Subject Classification 15A18.

A Cartan theorem for Banach algebras

Let A A be a semisimple Banach algebra, and let Ω A \Omega _A be its spectral unit ball. We show that every holomorphic map G : Ω A → Ω A G\colon \Omega _A\to \Omega _A satisfying G ( 0 ) = 0 G(0)=0 and G ′ ( 0 ) = I G’(0)=I fixes those elements of Ω A \Omega _A which belong to the centre of A A , but not necessarily any others. Using this, we deduce that the automorphisms of Ω A \Omega _A all leave the centre invariant. As a further application, we give a new proof of Nagasawa’s generalization of the Banach-Stone theorem.

Holomorphic Self-Maps of the Spectral unit Ball

Bulletin of the London Mathematical SocietyVolume 23, Issue 3 p. 256-262 Notes and papers Holomorphic Self-Maps of the Spectral unit Ball T. J. Ransford, T. J. Ransford Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SBSearch for more papers by this authorM. C. White, M. C. White Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SBSearch for more papers by this author T. J. Ransford, T. J. Ransford Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SBSearch for more papers by this authorM. C. White, M. C. White Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SBSearch for more papers by this author First published: May 1991 https://doi.org/10.1112/blms/23.3.256Citations: 14AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume23, Issue3May 1991Pages 256-262 RelatedInformation