Lecture 1 Covariant and Contravariant Calculi

United in the trinity functional calculus, spectrum, and spectral mapping theorem play the exceptional rôle in functional analysis and could not be substituted by anything else.

1.1 Functional Calculus as an Algebraic Homomorphism

Many traditional definitions of functional calculus are covered by the following rigid template based on the algebra homomorphism property:

The most typical definition of the spectrum is seemingly independent and uses the important notion of resolvent:

Then the following important theorem links spectrum and functional calculus together.

However the power of the classic spectral theory rapidly decreases if we move beyond the study of one normal operator (e.g. for quasinilpotent ones) and is virtually nil if we consider several non-commuting ones. Sometimes these severe limitations are seen to be irresistible and alternative constructions, i.e. model theory cf. Example 22 and [266], were developed.

Yet the spectral theory can be revived from a fresh start. While three compon­ents—functional calculus, spectrum, and spectral mapping theorem—are highly interdependent in various ways we will nevertheless arrange them as follows:

1. Functional calculus is an original notion defined in some independent terms;
2. Spectrum (or more specifically contravariant spectrum) (or spectral decomposition) is derived from previously defined functional calculus as its support (in some appropriate sense);
3. Spectral mapping theorem then should drop out naturally in the form (1) or some its variation.

Thus the entire scheme depends from the notion of the functional calculus and our ability to escape limitations of Definition 1. The first known to the present author definition of functional calculus not linked to algebra homomorphism property was the Weyl functional calculus defined by an integral formula [8]. Then its intertwining property with affine transformations of Euclidean space was proved as a theorem. However it seems to be the only “non-homomorphism” calculus for decades.

The different approach to whole range of calculi was given in [168] and developed in [173, 182, 180, 193] in terms of intertwining operators for group representations. 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.

In the following Subsections we will support the last claim by consideration of the simple known problem: characterisation a n × n matrix up to similarity. Even that “freshman” question can be only sorted out by the classical spectral theory for a small set of diagonalisable matrices. 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.

1.2 Intertwining Group Actions on Functions and Operators

Any functional calculus uses properties of functions to model properties of operators. Thus changing our viewpoint on functions, as was done in Section 5, we can get another approach to operators. The two main possibilities are encoded in Definitions 23 and 25: we can assign a certain function to the given operator or vice versa. Here we consider the second possibility and treat the first in the Subsection 1.5.

The representation 1 (23) is unitary irreducible when acts on the Hardy space H₂. Consequently we have one more reason to abolish the template definition 1: H₂ is not an algebra. 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.

The earliest functional calculus, which is not an algebraic homomorphism, was the Weyl functional calculus and was defined just by an integral formula as an operator valued distribution [8]. In that paper (joint) spectrum was defined as support of the Weyl calculus, i.e. as the set of point where this operator valued distribution does not vanish. We also define the spectrum as a support of functional calculus, but due to our Definition 4 it will means the set of non-vanishing intertwining operators with primary subrepresentations.

More variations of contravariant functional calculi are obtained from other groups and their representations [168, 173, 182, 180, 193].

A simple but important observation is that the Möbius transformations (1) can be easily extended to any Banach algebra. Let A be a Banach algebra with the unit e, an element a∈A with ||a||<1 be fixed, then for for transformations

we introduce SL₂(ℝ)-orbit

Clearly, A is a left (right) SL₂(ℝ)-homogeneous space with the natural left (right) action:

where g₁, g ∈ SL₂(ℝ) and b=g· a ∈ A. That is, for a fixed a∈A the set A and the SL₂(ℝ)-action on it are completely parametrised by correspondence b=g· a↔ g.

Let us define the resolvent function R(g,b): SL₂(ℝ) × A→ A by:

where

and

Then, we can directly check that:

for all g₁, g₂ ∈ SL₂(ℝ) and b ∈ A.

The last identity is well known in representation theory [159]*§ 13.2(10) and is a key ingredient of induced representations. Thus we can again linearise (2), cf. (23), in the space of continuous functions C(A,M) with values in a left A-module M:

where g ∈ SL₂(ℝ) and b ∈ A.

For any m∈ M we can define a K-invariant vacuum vector as v_m(g⁻¹· a)=m⊗ v₀(g⁻¹· a) ∈ C(A,M). It generates the associated with v_m family of coherent states v_m(u,a)=(ue−a)⁻¹m, where u∈ⅅ.

The wavelet transform defined by the same common formula based on coherent states (cf. (24)):

is a version of Cauchy integral, which maps L₂(A) to C(SL₂(ℝ),M). It is closely related (but not identical!) to the Riesz-Dunford functional calculus: the traditional functional calculus is given by the case:

The both conditions—the intertwining property and initial value—required by Definition 4 easily follows from our construction. Finally, we wish to provide an example of application of the Corollary 11.

1.3 Jet Bundles and Prolongations

Spectrum was defined in 5 as the support of our functional calculus. To elaborate its meaning we need the notion of a prolongation of representations introduced by S. Lie, see [267, 268] for a detailed exposition.

For a fixed n each holomorphic function f:ⅅ→ℂ has nth prolongation (or n-jet) j_nf: ⅅ → ℂⁿ⁺¹:

The graph amma_f⁽ⁿ⁾ of j_nf is a submanifold of Jⁿ which is section of the jet bundle over ⅅ with a fibre ℂⁿ⁺¹. We also introduce a notation J_n for the map J_n:f↦amma_f⁽ⁿ⁾ of a holomorphic f to the graph amma_f⁽ⁿ⁾ of its n-jet j_nf(z) (8).

One can prolong any map of functions ψ: f(z)↦ [ψ f](z) to a map ψ⁽ⁿ⁾ of n-jets by the formula

For example such a prolongation ρ₁⁽ⁿ⁾ of the representation ρ₁ of the group SL₂(ℝ) in H₂(ⅅ) (as any other representation of a Lie group [268]) will be again a representation of SL₂(ℝ). Equivalently we can say that J_n intertwines ρ₁ and ρ₁⁽ⁿ⁾:

Of course, the representation ρ₁⁽ⁿ⁾ is not irreducible: any jet subspace J^k, 0≤ k ≤ n is ρ₁⁽ⁿ⁾-invariant subspace of Jⁿ. However the representations ρ₁⁽ⁿ⁾ are primary [159]*§ 8.3 in the sense that they are not sums of two subrepresentations.

The following statement explains why jet spaces appeared in our study of functional calculus.

1.4 Spectrum and Spectral Mapping Theorem

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).

Recall the transitive on ⅅ group of inner automorphisms of SL₂(ℝ), which can send any λ∈ⅅ to 0 and are actually parametrised by such a λ. This group extends Proposition 8 to the complete characterisation of ρ_a for matrices.

Obviously this spectral theory is a fancy restatement of the Jordan normal form of matrices.

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(a)   (b)  (c)

Figure 1.1: Classical spectrum of the matrix from the Ex. 10 is shown at (a). Contravariant spectrum of the same matrix in the jet space is drawn at (b). The image of the contravariant spectrum under the map from Ex. 12 is presented at (c).

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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. The new version of spectrum is based on prolongation of ρ₁ into jet spaces (see Section ??). Naturally a correct version of spectral mapping theorem should also operate in jet spaces.

Let φ: ⅅ → ⅅ be a holomorphic map, let us define its action on functions [φ_* f](z)=f(φ(z)). According to the general formula (9) we can define the prolongation φ_*⁽ⁿ⁾ onto the jet space Jⁿ. Its associated action ρ₁^k φ_*⁽ⁿ⁾=φ_*⁽ⁿ⁾ρ₁ⁿ on the pairs (λ,k) is given by the formula:

where deg_λφ denotes the degree of zero of the function φ(z)−φ(λ) at the point z=λ and [x] denotes the integer part of x.

The explicit expression of (10) for φ_*⁽ⁿ⁾, which involves derivatives of φ upto nth order, is known, see for example [137]*Thm. 6.2.25, but was not recognised before as form of spectral mapping.

1.5 Functional Model and Spectral Distance

Let a be a matrix and µ(z) be its minimal polynomial:

If all eigenvalues λ_i of a (i.e. all roots of µ(z) belong to the unit disk we can consider the respective Blaschke product

such that its numerator coincides with the minimal polynomial µ(z). Moreover, for an unimodular z we have B_a(z)=µ_a(z)µ_a⁻¹(z)z^−m, where m=m₁+… +m_n. We also have the following covariance property:

The result follows from the observation that every elementary product z−λ _i/1−λ_iz is the Moebius transformation of z with the matrix (

). Thus the correspondence a↦ B_a(z) is a covariant (symbolic) calculus in the sense of the Defn. 23. See also the Example 22.

The Jordan normal form of a matrix provides a description, which is equivalent to its contravariant spectrum. From various viewpoints, e.g. numerical approximations, it is worth to consider its stability under a perturbation. 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:

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(a)  (b)

Figure 1.2: Perturbation of the Jordan block’s spectrum: (a) The spectrum of the perturbation J100+ε100 K of the Jordan block J100 by a random matrix K. (b) The spectrum of the random matrix K.

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The left picture in Fig. 1.2 presents a perturbation of a Jordan block J₁₀₀ by a random matrix. Perturbed eigenvalues are close to vertices of a right polygon with 100 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. 1.2). 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. We will describe a sense, in which the covariant spectrum of the matrix J_n+εⁿ K is stable for small ε. For this we introduce the covariant version of spectral distances motivated by the functional model. Our definition is different from other types known in the literature [324]*Ch. 5.

Since the spectral distance is defined through the distance in H₂ all standard axioms of a distance are automatically satisfied. For a Blaschke products we have | B_a(z) |=1 if | z |=1, thus ||B_a||_p=1 in any L_p on the unit circle. Therefore an alternative expression for the spectral distance is:

In particular, we always have 0≤ d(a,b) ≤ 2. We get an obvious consequence of Prop. 13, which justifies the name of the covariant spectral distance:

An important property of the covariant spectral distance is its stability under small perturbations.

In other words, a matrix with eigenvalues satisfying to the Lisdkii condition from the Thm. 14 is much closer to the Jordan block J₂ than a generic one with eigenvalues of the same order. Thus the covariant spectral distance is more stable under perturbation that magnitude of eigenvalues. For n=2 a proof can be forced through a direct calculation. We also conjecture that the similar statement is true for any n≥ 2.

1.6 Covariant Pencils of Operators

Let H be a real Hilbert space, possibly of finite dimensionality. For bounded linear operators A and B consider the generalised eigenvalue problem, that is finding a scalar λ and a vector x∈ H such that:

The standard eigenvalue problem corresponds to the case B=I, moreover for an invertible B the generalised problem can be reduced to the standard one for the operator B⁻¹A. Thus it is sensible to introduce the equivalence relation on the pairs of operators:

We may treat the pair (A,B) as a column vector (

). Then there is an action of the SL₂(ℝ) group on the pairs:

If we consider this SL₂(ℝ)-action subject to the equivalence relation (13) then we will arrive to a version of the linear-fractional transformation of the operator defined in (2). There is a connection of the SL₂(ℝ)-action (14) to the problem (12) through the following intertwining relation:

In other words the correspondence

is another realisation of a covariant calculus in the sense of Defn. 23. The collection of all pairs g· (A,B), g∈SL₂(ℝ) is an example of covariant pencil of operators. This set is a SL₂(ℝ)-homogeneous spaces, thus it shall be within the classification of such homogeneous spaces provided in the Subsection ??.

Hawing homogeneous spaces generated by pairs of operators we can define respective functions on those spaces. The special attention is due the following paraphrase of the resolvent:

Obviously R_(A,B)(g) contains the essential information about the pair (A,B). Probably, the function R_(A,B)(g) contains too much simultaneous information, we may restrict it to get a more detailed view. For vectors u, v∈ H we also consider vector and scalar-valued functions related to the generalised resolvent:

where (cA+dB)⁻¹u is understood as a solution w of the equation u=(cA+dB)w if it exists and is unique, this does not require the full invertibility of cA+dB.

It is easy to see that the map (A,B)↦ R_(A,B)^(u,v)(g) is a covariant calculus as well. It worth to notice that function R_(A,B) can again fall into three EPH cases.

Consider the space L(G) of functions spanned by all left translations of R_(A,B)(g). As usual, a closure in a suitable metric, say L_p, can be taken. The left action g: f(h)↦ f(g⁻¹h) of SL₂(ℝ) on this space is a linear representation of this group. Afterwards the representation can be decomposed into a sum of primary subrepresentations.

An important observation is that a decomposition into irreducible or primary components can reveal an EPH structure even in the cases hiding it on the homogeneous space level.

Let us briefly consider a quadratic eigenvalue problem: for given operators (matrices) A₀, A₁ and A₂ from B(H) find a scalar λ and a vector x∈ H such that

There is a connection with our study of conic sections from Subsection ?? which we will only hint for now. Comparing (16) with the equation of the cycle (3) we can associate the respective Fillmore–Springer–Cnops–type matrix to Q(λ), cf. (??):

Then we can state the following analogue of Thm. ?? for the quadratic eigenvalues:

So quadratic matrix polynomials are non-commuting analogues of the cycles and it would be exciting to extend the geometry from Section ?? to this non-commutative setting as much as possible.