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It was initially targeted for several non-commuting operators because no non-trivial algebra homomorphism is possible with a commutative algebra of function in this case. However it emerged later that the new definition is a useful replacement for classical one across all range of problems.

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Our solution in terms of new spectrum will be full and thus unavoidably coincides with one given by the Jordan normal form of matrices. Other more difficult questions are the subject of ongoing research. Any functional calculus uses properties of functions to model properties of operators. Instead we replace the homomorphism property by a symmetric covariance :. Note that our functional calculus released from the homomorphism condition can take value in any left A -module M , which however could be A itself if suitable.

This add much flexibility to our construction. In that paper joint spectrum was defined as support of the Weyl calculus, i. The wavelet transform defined by the same common formula based on coherent states cf. It is closely related but not identical! To elaborate its meaning we need the notion of a prolongation of representations introduced by S. Lie, see [ , ] for a detailed exposition.

Riesz-Fischer Theorem -- from Wolfram MathWorld

Now we are prepared to describe a spectrum of a matrix. Since the functional calculus is an intertwining operator its support is a decomposition into intertwining operators with primary representations we can not expect generally that these primary subrepresentations are irreducible.

Obviously this spectral theory is a fancy restatement of the Jordan normal form of matrices. As was mentioned in the beginning of this section a resonable spectrum should be linked to the corresponding functional calculus by an appropriate spectral mapping theorem.

Completing the Riesz-Dunford Functional Calculus

Naturally a correct version of spectral mapping theorem should also operate in jet spaces. However Fig. We also have the following covariance property:. The Jordan normal form of a matrix provides a description, which is equivalent to its contravariant spectrum. From various viewpoints, e. It is easy to see, that an arbitrarily small disturbance breaks the Jordan structure of a matrix. However, the result of random small perturbation will not be random, its nature is described by the following remarkable theorem:.

The left picture in Fig.

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Perturbed eigenvalues are close to vertices of a right polygon with vertices. Those regular arrangements occur despite of the fact that eigenvalues of the matrix K are dispersed through the unit disk the right picture in Fig.

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In a sense it is rather the Jordan block regularises eigenvalues of K than K perturbs the eigenvalue of the Jordan block. Although the Jordan structure itself is extremely fragile, it still can be easily guessed from a perturbed eigenvalues. Thus there exists a certain characterisation of matrices which is stable under small perturbations. For this we introduce the covariant version of spectral distances motivated by the functional model. Since the spectral distance is defined through the distance in H 2 all standard axioms of a distance are automatically satisfied.

Therefore an alternative expression for the spectral distance is:. We get an obvious consequence of Prop. An interesting twist is that one has to work with a weaker topology than the normthe so-called strict topology Buck, Rubel. A succinct introduction can be found in the wiki article "Contractions operator theory " and a complete account in "Harmonic Analysis of Operators in Hilbert Space" by Sz. Sign up to join this community.


The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 4 years, 4 months ago. Active 4 years, 3 months ago. Viewed times. Benjamin Benjamin 6 6 silver badges 19 19 bronze badges. Carlo Beenakker Carlo Beenakker Mike Jury Mike Jury 2, 11 11 silver badges 14 14 bronze badges.

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  • The application will take a million years to explain and shed no light on anything! I'm not sure if something like this can be applied to my context yet, that's why I left it open. So this is sort of a trivial analogue of Cayley-Hamilton.

    Completing the Riesz-Dunford Functional Calculus

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