aa r X i v : . [ m a t h . C V ] J un Erlangen Program at Large: Outline
Vladimir V. Kisil A BSTRACT . This is an outline of
Erlangen Program at Large . Study of objects andproperties, which are invariant under a group action, is very fruitful far beyondthe traditional geometry. In this paper we demonstrate this on the example ofthe group SL ( R ) . Starting from the conformal geometry we develop analyticfunctions and apply these to functional calculus. Finally we provide an extensivedescription of open problems. C ONTENTS
1. Introduction 12. Geometry 33. Analytic Functions 104. Functional Calculus 145. Open Problems 18References 20
1. Introduction
The simplest objects with non-commutative multiplication may be × ma-trices with real entries. Such matrices of determinant one form a closed set undermultiplication (since det ( AB ) = det A · det B ), the identity matrix is among themand any such matrix has an inverse (since det A = ). In other words those ma-trices form a group, the SL ( R ) group [32]—one of the two most important Liegroups in analysis. The other group is the Heisenberg group [8]. By contrastthe “ ax + b ”-group, which is often used to build wavelets, is only a subgroupof SL ( R ) , see the numerator in (1.1).The simplest non-linear transforms of the real line—linear-fractional or M ¨obiusmaps—may also be associated with × matrices [4, Ch. 13]:(1.1) g : x g · x = ax + bcx + d , where g = (cid:18) a bc d (cid:19) , x ∈ R . Mathematics Subject Classification.
Primary 30G35; Secondary 22E46, 30F45, 32F45, 43A85,30G30, 42C40, 46H30, 47A13, 81R30, 81R60.
Key words and phrases.
Special linear group, Hardy space, Clifford algebra, elliptic, parabolic,hyperbolic, complex numbers, dual numbers, double numbers, split-complex numbers, Cauchy-Riemann-Dirac operator, M¨obius transformations, functional calculus, spectrum, quantum mechanics,non-commutative geometry.On leave from the Odessa University.
An enjoyable calculation shows that the composition of two transforms (1.1) withdifferent matrices g and g is again a M ¨obius transform with matrix the product g g . In other words (1.1) it is a (left) action of SL ( R ) .According to F. Klein’s Erlangen program (which was influenced by S. Lie) anygeometry is dealing with invariant properties under a certain group action. Forexample, we may ask:
What kinds of geometry are related to the SL ( R ) action (1.1)?The Erlangen program has probably the highest rate of praisedactually used among math-ematical theories not only due to the big numerator but also due to undeservingsmall denominator. As we shall see below Klein’s approach provides some sur-prising conclusions even for such over-studied objects as circles. It is easy to see that the SL ( R ) ac-tion (1.1) makes sense also as a map of complex numbers z = x + i y , i = − .Moreover, if y > then g · z has a positive imaginary part as well, i.e. (1.1) definesa map from the upper half-plane to itself.However there is no need to be restricted to the traditional route of complexnumbers only. Less-known dual and double numbers [37, Suppl. C] have also theform z = x + i y but different assumptions on the imaginary unit i : i = or i = correspondingly. Although the arithmetic of dual and double numbers is differentfrom the complex ones, e.g. they have divisors of zero, we are still able to definetheir transforms by (1.1) in most cases.Three possible values − , and of σ := i will be refereed to here as el-liptic , parabolic and hyperbolic cases respectively. We repeatedly meet such a divi-sion of various mathematical objects into three classes. They are named by thehistorically first example—the classification of conic sections—however the pat-tern persistently reproduces itself in many different areas: equations, quadraticforms, metrics, manifolds, operators, etc. We will abbreviate this separation as EPH-classification . The common origin of this fundamental division can be seenfrom the simple picture of a coordinate line split by zero into negative and pos-itive half-axes:(1.2) +− ↑ parabolic elliptichyperbolic Connections between different objects admitting EPH-classification are notlimited to this common source. There are many deep results linking, for exam-ple, ellipticity of quadratic forms, metrics and operators. On the other hand thereare still a lot of white spots and obscure gaps between some subjects as well.To understand the action (1.1) in all EPH cases we use the Iwasawa decom-position [32] of SL ( R ) = ANK into three one-dimensional subgroups A , N and K :(1.3) (cid:18) a bc d (cid:19) = (cid:18) α α − (cid:19)(cid:18) ν (cid:19)(cid:18) cos φ sin φ − sin φ cos φ (cid:19) . Subgroups A and N act in (1.1) irrespectively to value of σ : A makes a dilation by α , i.e. z α z , and N shifts points to left by ν , i.e. z z + ν .By contrast, the action of the third matrix from the subgroup K sharply de-pends on σ , see Fig. 1. In elliptic, parabolic and hyperbolic cases K -orbits are cir-cles, parabolas and (equilateral) hyperbolas correspondingly. Thin traversal linesin Fig. 1 join points of orbits for the same values of φ and grey arrows represent“local velocities”—vector fields of derived representations. RLANGEN PROGRAM AT LARGE: OUTLINE 3 F IGURE Action of the K subgroup. The corresponding K -orbits arethick circles, parabolas and hyperbolas. Thin traversal lines are imagesof the vertical axis for certain values of the parameter φ . As we already mentioned the division ofmathematics into areas is only apparent. Therefore it is unnatural to limit Erlan-gen program only to “geometry”. We may continue to look for SL ( R ) invariantobjects in other related fields. For example, transform (1.1) generates unitary rep-resentations on certain L spaces, cf. (1.1):(1.4) g − : f ( x ) ( cx + d ) m f (cid:18) ax + bcx + d (cid:19) . For m = , , . . . the invariant subspaces of L are Hardy and (weighted)Bergman spaces of complex analytic functions. All main objects of complex analysis (Cauchy and Bergman integrals, Cauchy-Riemann and Laplace equations, Taylorseries etc.) may be obtaining in terms of invariants of the discrete series represen-tations of SL ( R ) [20, § principal and complimen-tary [32]) play the similar r ˆoles for hyperbolic and parabolic cases [20, 24].Moving further we may observe that transform (1.1) is defined also for anelement x in any algebra A with a unit as soon as ( cx + d ) ∈ A has an inverse.If A is equipped with a topology, e.g. is a Banach algebra, then we may study a functional calculus for element x [23] in this way. It is defined as an intertwiningoperator between the representation (1.4) in a space of analytic functions and asimilar representation in a left A -module.In the spirit of Erlangen program such functional calculus is still a geometry,since it is dealing with invariant properties under a group action. However evenfor a simplest non-normal operator, e.g. a Jordan block of the length k , the ob-tained space is not like a space of point but is rather a space of k -th jets [23]. Suchnon-point behaviour is oftenly attributed to non-commutative geometry and Erlan-gen program provides an important input on this fashionable topic [20].Of course, there is no reasons to limit Erlangen program to SL ( R ) group only,other groups may be more suitable in different situations. However SL ( R ) stillpossesses a big unexplored potential and is a good object to start with.
2. Geometry2.1. Cycles as Invariant Objects.Definition 2.1.
The common name cycle [37] is used to denote circles, parabolasand hyperbolas (as well as straight lines as their limits) in the respective EPH case.It is well known that any cycle is a conic sections and an interesting observationis that corresponding K -orbits are in fact sections of the same two-sided right-anglecone, see Fig. 2. Moreover, each straight line generating the cone, see Fig. 2(b), iscrossing corresponding EPH K -orbits at points with the same value of parameter φ from (1.3). In other words, all three types of orbits are generated by the rotationsof this generator along the cone. VLADIMIR V. KISIL (a)
E E ′ P P ′ HH ′ (b) E E ′ P P ′ HH ′ F IGURE K -orbits as conic sections: circles are sections by the plane EE ′ ; parabolas are sections by PP ′ ; hyperbolas are sections by HH ′ . Pointson the same generator of the cone correspond to the same value of φ . K -orbits are K -invariant in a trivial way. Moreover since actions of both A and N for any σ are extremely “shape-preserving” we find natural invariant objects ofthe M ¨obius map: Theorem 2.2 ([27]) . The family of all cycles from Defn. 2.1 is invariant under the ac-tion (1.1) . According to Erlangen ideology we shall study invariant properties of cycles.
Fig. 2 suggests that we may get a unified treatmentof cycles in all EPH by consideration of a higher dimension spaces. The standardmathematical method is to declare objects under investigations (cycles in our case,functions in functional analysis, etc.) to be simply points of some bigger space.This space should be equipped with an appropriate structure to hold externallyinformation which were previously inner properties of our objects.A generic cycle is the set of points ( u , v ) ∈ R defined for all values of σ by theequation(2.1) k ( u − σv ) − lu − nv + m = This equation (and the corresponding cycle) is defined by a point ( k , l , n , m ) froma projective space P , since for a scaling factor λ = the point ( λk , λl , λn , λm ) defines the same equation (2.1). We call P the cycle space and refer to the initial R as the point space .In order to get a connection with M ¨obius action (1.1) we arrange numbers ( k , l , n , m ) into the matrix(2.2) C s ˘ σ = (cid:18) l + ˘ı sn − mk − l + ˘ı sn (cid:19) , with a new imaginary unit ˘ı and an additional parameter s usually equal to ± .The values of ˘ σ := ˘ı is − , or independently from the value of σ . The ma-trix (2.2) is the cornerstone of (extended) Fillmore–Springer–Cnops construction(FSCc) [5] and closely related to technique recently used by A.A. Kirillov to studythe Apollonian gasket [12].The significance of FSCc in Erlangen framework is provided by the followingresult: Theorem 2.3.
The image ˜ C s ˘ σ of a cycle C s ˘ σ under transformation (1.1) with g ∈ SL ( R ) is given by similarity of the matrix (2.2) : (2.3) ˜ C s ˘ σ = gC s ˘ σ g − . RLANGEN PROGRAM AT LARGE: OUTLINE 5
In other words FSCc (2.2) intertwines
M¨obius action (1.1) on cycles with linear map (2.3) . There are several ways to prove (2.3): either by a brute force calculation (fortu-nately performed by a CAS) [24] or through the related orthogonality of cycles [5],see the end of the next section 2.3.The important observation here is that FSCc (2.2) uses an imaginary unit ˘ı which is not related to i defining the appearance of cycles on plane. In other wordsany EPH type of geometry in the cycle space P admits drawing of cycles in thepoint space R as circles, parabolas or hyperbolas. We may think on points of P as ideal cycles while their depictions on R are only their shadows on the wall ofPlato’s cave.(a) c e c p c h r c e c p c h r (b) c e f e f p f h c e f e f p f h F IGURE (a) Different EPH implementations of the same cycles de-fined by quadruples of numbers.(b) Centres and foci of two parabolas with the same focal length. Fig. 3(a) shows the same cycles drawn in different EPH styles. Points c e , p , h =( lk , − σ nk ) are their respective e/p/h-centres. They are related to each other throughseveral identities:(2.4) c e = ¯ c h , c p = ( c e + c h ) . Fig. 3(b) presents two cycles drawn as parabolas, they have the same focal length n k and thus their e-centres are on the same level. In other words concentric parabo-las are obtained by a vertical shift, not scaling as an analogy with circles or hyper-bolas may suggest.Fig. 3(b) also presents points, called e/p/h-foci:(2.5) f e , p , h = (cid:18) lk , − det C s ˘ σ nk (cid:19) , which are independent of the sign of s . If a cycle is depicted as a parabola thenh-focus, p-focus, e-focus are correspondingly geometrical focus of the parabola, itsvertex, and the point on the directrix nearest to the vertex.As we will see, cf. Thms. 2.5 and 2.7, all three centres and three foci are usefulattributes of a cycle even if it is drawn as a circle. We use known algebraic invariantsof matrices to build appropriate geometric invariants of cycles. It is yet anotherdemonstration that any division of mathematics into subjects is only illusive.For × matrices (and thus cycles) there are only two essentially differentinvariants under similarity (2.3) (and thus under M ¨obius action (1.1)): the trace and the determinant . The latter was already used in (2.5) to define cycle’s foci.However due to projective nature of the cycle space P the absolute values of traceor determinant are irrelevant, unless they are zero. VLADIMIR V. KISIL
Alternatively we may have a special arrangement for normalisation of quadru-ples ( k , l , n , m ) . For example, if k = we may normalise the quadruple to ( lk , nk , mk ) with highlighted cycle’s centre. Moreover in this case det C s ˘ σ is equal to the squareof cycle’s radius, cf. Section 2.6. Another normalisation det C s ˘ σ = is used in [12]to get a nice condition for touching circles.We still get important characterisation even with non-normalised cycles, e.g.,invariant classes (for different ˘ σ ) of cycles are defined by the condition det C s ˘ σ = .Such a class is parametrises only by two real number and as such is easily attachedto certain point of R . For example, the cycle C s ˘ σ with det C s ˘ σ = , ˘ σ = − drawnelliptically represent just a point ( lk , nk ) , i.e. (elliptic) zero-radius circle. The samecondition with ˘ σ = in hyperbolic drawing produces a null-cone originated atpoint ( lk , nk ) : ( u − lk ) − ( v − nk ) = i.e. a zero-radius cycle in hyperbolic metric.F IGURE Different i -implementations of the same ˘ σ -zero-radius cyclesand corresponding foci. In general for every notion there is nine possibilities: three EPH cases in thecycle space times three EPH realisations in the point space. Such nine cases for“zero radius” cycles is shown on Fig. 4. For example, p-zero-radius cycles in anyimplementation touch the real axis.This “touching” property is a manifestation of the boundary effect in the upper-half plane geometry [24, Rem. 3.4]. The famous question on hearing drum’s shapehas a sister:
Can we see/feel the boundary from inside a domain?
Both orthogonality relations described below are “boundary aware” as well. It isnot surprising after all since SL ( R ) action on the upper-half plane was obtainedas an extension of its action (1.1) on the boundary.According to the categorical viewpoint internal properties of objects are of mi-nor importance in comparison to their relations with other objects from the sameclass. Thus from now on we will look for invariant relations between two or morecycles. The most expected relation between cy-cles is based on the following M ¨obius invariant “inner product” build from a traceof product of two cycles as matrices:(2.6) D C s ˘ σ , ˜ C s ˘ σ E = tr ( C s ˘ σ ˜ C s ˘ σ ) By the way, an inner product of this type is used, for example, in GNS constructionto make a Hilbert space out of C ∗ -algebra. The next standard move is given by thefollowing definition. RLANGEN PROGRAM AT LARGE: OUTLINE 7
Definition 2.4.
Two cycles are called ˘ σ -orthogonal if D C s ˘ σ , ˜ C s ˘ σ E = .For the case of ˘ σσ = , i.e. when geometries of the cycle and point spaces areboth either elliptic or hyperbolic, such an orthogonality is the standard one, de-fined in terms of angles between tangent lines in the intersection points of two cy-cles. However in the remaining seven ( = − ) cases the innocent-looking Defn. 2.4brings unexpected relations. a bc dσ = −
1, ˘ σ = − a bc dσ = −
1, ˘ σ = a bc dσ = −
1, ˘ σ = F IGURE Orthogonality of the first kind in the elliptic point space.Each picture presents two groups (green and blue) of cycles which areorthogonal to the red cycle C s ˘ σ . Point b belongs to C s ˘ σ and the family ofblue cycles passing through b is orthogonal to C s ˘ σ . They all also intersectin the point d which is the inverse of b in C s ˘ σ . Any orthogonality is re-duced to the usual orthogonality with a new (“ghost”) cycle (shown bythe dashed line), which may or may not coincide with C s ˘ σ . For any point a on the “ghost” cycle the orthogonality is reduced to the local notion inthe terms of tangent lines at the intersection point. Consequently such apoint a is always the inverse of itself. Elliptic (in the point space) realisations of Defn. 2.4, i.e. σ = − is shown inFig. 5. The left picture corresponds to the elliptic cycle space, e.g. ˘ σ = − . Theorthogonality between the red circle and any circle from the blue or green familiesis given in the usual Euclidean sense. The central (parabolic in the cycle space) andthe right (hyperbolic) pictures show non-local nature of the orthogonality. Thereare analogues pictures in parabolic and hyperbolic point spaces as well [24].This orthogonality may still be expressed in the traditional sense if we willassociate to the red circle the corresponding “ghost” circle, which shown by thedashed line in Fig. 5. To describe ghost cycle we need the Heaviside function χ ( σ ) :(2.7) χ ( t ) = (cid:12) t > − t < Theorem 2.5.
A cycle is ˘ σ -orthogonal to cycle C s ˘ σ if it is orthogonal in the usual sense tothe σ -realisation of “ghost” cycle ˆ C s ˘ σ , which is defined by the following two conditions: (i) χ ( σ ) -centre of ˆ C s ˘ σ coincides with ˘ σ -centre of C s ˘ σ . (ii) Cycles ˆ C s ˘ σ and C s ˘ σ have the same roots, moreover det ˆ C σ = det C χ ( ˘ σ ) σ . The above connection between various centres of cycles illustrates their mean-ingfulness within our approach.One can easy check the following orthogonality properties of the zero-radiuscycles defined in the previous section:(i) Since (cid:10) C s ˘ σ , C s ˘ σ (cid:11) = det C s ˘ σ zero-radius cycles are self-orthogonal (isotropic)ones.(ii) A cycle C s ˘ σ is σ -orthogonal to a zero-radius cycle Z s ˘ σ if and only if C s ˘ σ passes through the σ -centre of Z s ˘ σ . VLADIMIR V. KISIL
With appetite alreadywet one may wish to build more joint invariants. Indeed for any homogeneouspolynomial p ( x , x , . . . , x n ) of several non-commuting variables one may definean invariant joint disposition of n cycles j C s ˘ σ by the condition: tr p ( C s ˘ σ , C s ˘ σ , . . . , n C s ˘ σ ) = However it is preferable to keep some geometrical meaning of constructed no-tions.An interesting observation is that in the matrix similarity of cycles (2.3) onemay replace element g ∈ SL ( R ) by an arbitrary matrix corresponding to anothercycle. More precisely the product C s ˘ σ ˜ C s ˘ σ C s ˘ σ is again the matrix of the form (2.2) andthus may be associated to a cycle. This cycle may be considered as the reflectionof ˜ C s ˘ σ in C s ˘ σ . Definition 2.6.
A cycle C s ˘ σ is s-orthogonal to a cycle ˜ C s ˘ σ if the reflection of ˜ C s ˘ σ in C s ˘ σ is orthogonal (in the sense of Defn. 2.4) to the real line. Analytically this is definedby:(2.8) tr ( C s ˘ σ ˜ C s ˘ σ C s ˘ σ R s ˘ σ ) = Due to invariance of all components in the above definition s-orthogonalityis a M ¨obius invariant condition. Clearly this is not a symmetric relation: if C s ˘ σ iss-orthogonal to ˜ C s ˘ σ then ˜ C s ˘ σ is not necessarily s-orthogonal to C s ˘ σ . a bc dσ = −
1, ˘ σ = − a bc dσ = −
1, ˘ σ = a bc dσ = −
1, ˘ σ = F IGURE Orthogonality of the second kind for circles. To highlightboth similarities and distinctions with the ordinary orthogonality we usethe same notations as that in Fig. 5.
Fig. 6 illustrates s-orthogonality in the elliptic point space. By contrast withFig. 5 it is not a local notion at the intersection points of cycles for all ˘ σ . Howeverit may be again clarified in terms of the appropriate s-ghost cycle, cf. Thm. 2.5. Theorem 2.7.
A cycle is s-orthogonal to a cycle C s ˘ σ if its orthogonal in the traditionalsense to its s-ghost cycle ˜ C ˘ σ ˘ σ = C χ ( σ ) ˘ σ R ˘ σ ˘ σ C χ ( σ ) ˘ σ , which is the reflection of the real line in C χ ( σ ) ˘ σ and χ is the Heaviside function (2.7) . Moreover (i) χ ( σ ) -Centre of ˜ C ˘ σ ˘ σ coincides with the ˘ σ -focus of C s ˘ σ , consequently all lines s-orthogonal to C s ˘ σ are passing the respective focus. (ii) Cycles C s ˘ σ and ˜ C ˘ σ ˘ σ have the same roots. Note the above intriguing interplay between cycle’s centres and foci. Al-though s-orthogonality may look exotic it will naturally appear in the end of nextSection again.Of course, it is possible to define another interesting higher order joint invari-ants of two or even more cycles.
RLANGEN PROGRAM AT LARGE: OUTLINE 9 (a) z z z z (b) z z z z d e d p (c) A B ε ~ CD F IGURE (a) The square of the parabolic diameter is the square of thedistance between roots if they are real ( z and z ), otherwise the negativesquare of the distance between the adjoint roots ( z and z ).(b) Distance as extremum of diameters in elliptic ( z and z ) and parabolic( z and z ) cases.(c) Perpendicular as the shortest route to a line. Geo metry in the plain meaningof this word deals with distances and lengths . Can we obtain them from cycles?We mentioned already that for circles normalised by the condition k = thevalue det C s ˘ σ = (cid:10) C s ˘ σ , C s ˘ σ (cid:11) produces the square of the traditional circle radius. Thuswe may keep it as the definition of the radius for any cycle. But then we needto accept that in the parabolic case the radius is the (Euclidean) distance between(real) roots of the parabola, see Fig. 7(a).Having radii of circles already defined we may use them for other measure-ments in several different ways. For example, the following variational definitionmay be used: Definition 2.8.
The distance between two points is the extremum of diameters ofall cycles passing through both points, see Fig. 7(b).If ˘ σ = σ this definition gives in all EPH cases the distance between endpointsof a vector z = u + i v as follows:(2.9) d e , p , h ( u , v ) = ( u + i v )( u − i v ) = u − σv . The parabolic distance d p = u , see Fig. 7(b), algebraically sits between d e and d h according to the general principle (1.2) and is widely accepted [37]. However onemay be unsatisfied by its degeneracy.An alternative measurement is motivated by the fact that a circle is the set ofequidistant points from its centre. However the choice of “centre” is now rich: itmay be either point from three centres (2.4) or three foci (2.5). Definition 2.9.
The length of a directed interval −→ AB is the radius of the cycle withits centre (denoted by l c ( −→ AB ) ) or focus (denoted by l f ( −→ AB ) ) at the point A whichpasses through B .These definition is less common and have some unusual properties like non-symmetry: l f ( −→ AB ) = l f ( −→ BA ) . However it comfortably fits the Erlangen programdue to its SL ( R ) - conformal invariance : Theorem 2.10 ([24]) . Let l denote either the EPH distances (2.9) or any length fromDefn. 2.9. Then for fixed y , y ′ ∈ R σ the limit: lim t → l ( g · y , g · ( y + ty ′ )) l ( y , y + ty ′ ) , where g ∈ SL ( R ) , exists and its value depends only from y and g and is independent from y ′ . We may return from distances to angles recalling that in the Euclidean space aperpendicular provides the shortest root from a point to a line, see Fig. 7(c).
Definition 2.11.
Let l be a length or distance. We say that a vector −→ AB is l -perpendicular to a vector −→ CD if function l ( −→ AB + ε −→ CD ) of a variable ε has a localextremum at ε = .A pleasant surprise is that l f -perpendicularity obtained thought the lengthfrom focus (Defn. 2.9) coincides with already defined in Section 2.5 s-orthogonalityas follows from Thm. 2.7(i). It is also possible [13] to make SL ( R ) action isometricin all three cases.All these study are waiting to be generalised to high dimensions and Cliffordalgebras provide a suitable language for this [24].
3. Analytic Functions
We saw in the previous section that an inspiring geometry of cycles can berecovered from the properties of SL ( R ) . In this section we consider a realisationof the function theory within Erlangen approach [16, 17, 19, 20]. Elements of SL ( R ) could bealso represented by × -matrices with complex entries such that: g = (cid:18) α ¯ ββ ¯ α (cid:19) , g − = (cid:18) ¯ α − ¯ β − β α (cid:19) , | α | − | β | = This realisations of SL ( R ) (or rather SU ( C ) ) is more suitable for function theoryin the unit disk. It is obtained from the form, which we used before for the upperhalf-plane, by means of the Cayley transform [24, § D with the homogeneous space SL ( R ) / T for theunit circle T through the important decomposition SL ( R ) ∼ D × T with K = T —the only compact subgroup of SL ( R ) : (cid:18) α ¯ ββ ¯ α (cid:19) = | α | (cid:18) β ¯ α − βα − (cid:19) α | α | ¯ α | α | ! = q − | u | (cid:18) u ¯ u (cid:19) , (cid:18) e iω e − iω (cid:19) where ω = arg α , u = ¯ β ¯ α − , | u | < Each element g ∈ SL ( R ) acts by the linear-fractional transformation (the M ¨obiusmap) on D and T H ( T ) as follows:(3.1) g − : z ¯ αz − ¯ βα − βz , where g − = (cid:18) ¯ α − ¯ β − β α (cid:19) . In the decomposition (3.1) the first matrix on the right hand side acts by transfor-mation (1.1) as an orthogonal rotation of T or D ; and the second one—by transitivefamily of maps of the unit disk onto itself.The standard linearisation procedure [10, § ρ irreducible on the Hardy space :(3.2) ρ ( g ) : f ( z ) α − βz f (cid:18) ¯ αz − ¯ βα − βz (cid:19) where g − = (cid:18) ¯ α − ¯ β − β α (cid:19) . RLANGEN PROGRAM AT LARGE: OUTLINE 11
M ¨obius transformations provide a natural family of intertwining operators for ρ coming from inner automorphisms of SL ( R ) (will be used later).We choose [18, 19] K -invariant function v ( z ) ≡ to be a vacuum vector . Thusthe associated coherent states v ( g , z ) = ρ ( g ) v ( z ) = ( u − z ) − are completely determined by the point on the unit disk u = ¯ β ¯ α − . The family ofcoherent states considered as a function of both u and z is obviously the Cauchykernel [16]. The wavelet transform [16, 18] W : L ( T ) → H ( D ) : f ( z ) W f ( g ) = h f , v g i is the Cauchy integral :(3.3) W f ( u ) = πi Z T f ( z ) u − z dz . We start from the following observation reflected in the almost any textbookon complex analysis:
Proposition 3.1.
Analytic function theory in the unit disk D is a manifestation of themock discrete series representation ρ of SL ( R ) : (3.4) ρ ( g ) : f ( z ) α − βz f (cid:18) ¯ αz − ¯ βα − βz (cid:19) , where (cid:18) ¯ α − ¯ β − β α (cid:19) ∈ SL ( R ) . Other classical objects of complex analysis (the Cauchy-Riemann equation, theTaylor series, the Bergman space, etc.) can be also obtained [16, 19] from represen-tation ρ as shown below. Consideration ofLie groups is hardly possible without consideration of their Lie algebras, whichare naturally represented by left and right invariant vectors fields on groups. Ona homogeneous space Ω = G/H we have also defined a left action of G and canbe interested in left invariant vector fields (first order differential operators). Dueto the irreducibility of F ( Ω ) under left action of G every such vector field D re-stricted to F ( Ω ) is a scalar multiplier of identity D | F ( Ω ) = cI . We are in particularinterested in the case c = . Definition 3.2. [2, 31] A G -invariant first order differential operator D τ : C ∞ ( Ω , S ⊗ V τ ) → C ∞ ( Ω , S ⊗ V τ ) such that W ( F ( X )) ⊂ ker D τ is called (Cauchy-Riemann-)Dirac operator on Ω = G/H associated with an irreducible representation τ of H in a space V τ and a spinorbundle S .The Dirac operator is explicitly defined by the formula [31, (3.1)]:(3.5) D τ = n X j = ρ ( Y j ) ⊗ c ( Y j ) ⊗ where Y j is an orthonormal basis of p = h ⊥ —the orthogonal completion of the Liealgebra h of the subgroup H in the Lie algebra g of G ; ρ ( Y j ) is the infinitesimalgenerator of the right action of G on Ω ; c ( Y j ) is Clifford multiplication by Y i ∈ p on the Clifford module S . We also define an invariant Laplacian by the formula(3.6) ∆ τ = n X j = ρ ( Y j ) ⊗ ǫ j ⊗ where ǫ j = c ( Y j ) is + or − . Proposition 3.3.
Let all commutators of vectors of h ⊥ belong to h , i.e. [h ⊥ , h ⊥ ] ⊂ h . Letalso f be an eigenfunction for all vectors of h with eigenvalue and let also W f be a nullsolution to the Dirac operator D . Then ∆f ( x ) = for all f ( x ) ∈ F ( Ω ) . P ROOF . Because ∆ is a linear operator and F ( Ω ) is generated by π ( s ( a )) W f it is enough to check that ∆π ( s ( a )) W f = . Because ∆ and π commute it isenough to check that ∆ W f = . Now we observe that ∆ = D − X i , j ρ ([ Y i , Y j ]) ⊗ c ( Y i ) c ( Y j ) ⊗ Thus the desired assertion is follows from two identities ρ ([ Y i , Y j ]) W f = for [ Y i , Y j ] ∈ H and D W f = . (cid:3) Example 3.4.
Let G = SL ( R ) and H be its one-dimensional compact subgroup K generated by an element Z ∈ sl( R ) . Then h ⊥ is spanned by two vectors Y = A and Y = B . In such a situation we can use C instead of the Clifford algebra.Then formula (3.5) takes a simple form D = r ( A + iB ) . Infinitesimal action of thisoperator in the upper-half plane follows from calculation in [32, VI.5(8), IX.5(3)],it is [ D H f ]( z ) = − iy ∂f ( z ) ∂ ¯ z , z = x + iy . Making the Caley transform we can find itsaction in the unit disk D D : again the Cauchy-Riemann operator ∂∂ ¯ z is its principalcomponent. We calculate D H explicitly now to stress the similarity with R case.For the upper half plane H we have following formulas: s : H → SL ( R ) : z = x + iy g = (cid:18) y / xy − / y − / (cid:19) ; s − : SL ( R ) → H : (cid:18) a bc d (cid:19) z = ai + bci + d ; ρ ( g ) : H → H : z s − ( s ( z ) ∗ g )= s − (cid:18) ay − / + cxy − / by / + dxy − / cy − / dy − / (cid:19) = ( yb + xd ) + i ( ay + cx ) ci + d Thus the right action of SL ( R ) on H is given by the formula ρ ( g ) z = ( yb + xd ) + i ( ay + cx ) ci + d = x + y bd + acc + d + iy c + d . For A and B in sl( R ) we have: ρ ( e At ) z = x + iye t , ρ ( e Bt ) z = x + y e t − e − t e t + e − t + iy e t + e − t . Thus [ ρ ( A ) f ]( z ) = ∂f ( ρ ( e At ) z ) ∂t | t = = y∂ f ( z ) , [ ρ ( B ) f ]( z ) = ∂f ( ρ ( e Bt ) z ) ∂t | t = = y∂ f ( z ) , where ∂ and ∂ are derivatives of f ( z ) with respect to real and imaginary party of z respectively. Thus we get D H = iρ ( A ) + ρ ( B ) = yi∂ + y∂ = y ∂∂ ¯ z as was expected. RLANGEN PROGRAM AT LARGE: OUTLINE 13
For any decomposition f a ( x ) = P α ψ α ( x ) V α ( a ) of the coherent states f a ( x ) by means of functions V α ( a ) (where the sum can be-come eventually an integral) we have the Taylor expansion b f ( a ) = Z X f ( x ) ¯ f a ( x ) dx = Z X f ( x ) X α ¯ ψ α ( x ) ¯ V α ( a ) dx = X α Z X f ( x ) ¯ ψ α ( x ) dx ¯ V α ( a )= ∞ X α ¯ V α ( a ) f α , (3.7)where f α = R X f ( x ) ¯ ψ α ( x ) dx . However to be useful within the presented schemesuch a decomposition should be connected with the structures of G , H , and therepresentation π . We will use a decomposition of f a ( x ) by the eigenfunctions ofthe operators π ( h ) , h ∈ h . Definition 3.5.
Let F = R A H α dα be a spectral decomposition with respect to theoperators π ( h ) , h ∈ h . Then the decomposition(3.8) f a ( x ) = Z A V α ( a ) f α ( x ) dα , where f α ( x ) ∈ H α and V α ( a ) : H α → H α is called the Taylor decomposition of theCauchy kernel f a ( x ) .Note that the Dirac operator D is defined in the terms of left invariant shiftsand therefor commutes with all π ( h ) . Thus it also has a spectral decompositionover spectral subspaces of π ( h ) :(3.9) D = Z A D δ dδ . We have obvious property
Proposition 3.6.
If spectral measures dα and dδ from (3.8) and (3.9) have disjoint sup-ports then the image of the Cauchy integral belongs to the kernel of the Dirac operator. For discrete series representation functions f α ( x ) can be found in F (as in Ex-ample 3.7), for the principal series representation this is not the case. To overcomeconfusion one can think about the Fourier transform on the real line. It can beregarded as a continuous decomposition of a function f ( x ) ∈ L ( R ) over a set ofharmonics e iξx neither of those belongs to L ( R ) . This has a lot of common withthe Example 3.10(b) in [16]. Example 3.7.
Let G = SL ( R ) and H = K be its maximal compact subgroup and π defined in (3.2). H acts on T by rotations. It is one dimensional and eigenfunc-tions of its generator Z are parametrized by integers (due to compactness of K ).Moreover, on the irreducible Hardy space these are positive integers n =
1, 2, 3 . . . and corresponding eigenfunctions are f n ( φ ) = e i ( n − ) φ . Negative integers spanthe space of anti-holomorphic function and the splitting reflects the existence ofanalytic structure given by the Cauchy-Riemann equation. The decomposition ofcoherent states f a ( φ ) by means of this functions is well known: f a ( φ ) = q − | a | ¯ ae iφ − = ∞ X n = q − | a | ¯ a n − e i ( n − ) φ = ∞ X n = V n ( a ) f n ( φ ) , where V n ( a ) = q − | a | ¯ a n − . This is the classical Taylor expansion up to multi-pliers coming from the invariant measure.
4. Functional Calculus
United in the trinity functional calculus, spectrum, and spectral mapping the-orem play the exceptional r ˆole in functional analysis and could not be substitutedby anything else. All traditional definitions of functional calculus are covered bythe following rigid template based on algebra homomorphism property:
Definition 4.1. An functional calculus for an element a ∈ A is a continuous linearmapping Φ : A → A such that(i) Φ is a unital algebra homomorphism Φ ( f · g ) = Φ ( f ) · Φ ( g ) . (ii) There is an initialisation condition: Φ [ v ] = a for for a fixed function v ,e.g. v ( z ) = z .Most typical definition of the spectrum is seemingly independent and uses theimportant notion of resolvent: Definition 4.2. A resolvent of element a ∈ A is the function R ( λ ) = ( a − λe ) − ,which is the image under Φ of the Cauchy kernel ( z − λ ) − .A spectrum of a ∈ A is the set sp a of singular points of its resolvent R ( λ ) .Then the following important theorem links spectrum and functional calculustogether. Theorem 4.3 (Spectral Mapping) . For a function f suitable for the functional calculus: (4.1) f ( sp a ) = sp f ( a ) . However the power of the classic spectral theory rapidly decreases if we movebeyond the study of one normal operator (e.g. for quasinilpotent ones) and is vir-tually nil if we consider several non-commuting ones. Sometimes these severelimitations are seen to be irresistible and alternative constructions, i.e. model the-ory [33], were developed.Yet the spectral theory can be revived from a fresh start. While three compon-ents—functional calculus, spectrum, and spectral mapping theorem—are highlyinterdependent in various ways we will nevertheless arrange them as follows:(i) Functional calculus is an original notion defined in some independentterms;(ii) Spectrum (or spectral decomposition) is derived from previously de-fined functional calculus as its support (in some appropriate sense);(iii) Spectral mapping theorem then should drop out naturally in the form (4.1)or some its variation.Thus the entire scheme depends from the notion of the functional calculusand our ability to escape limitations of Definition 4.1. The first known to thepresent author definition of functional calculus not linked to algebra homomor-phism property was the Weyl functional calculus defined by an integral formula [1].Then its intertwining property with affine transformations of Euclidean space wasproved 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 [14] and devel-oped in [18] in terms of intertwining operators for group representations. It wasinitially targeted for several non-commuting operators because no non-trivial al-gebra homomorphism with a commutative algebra of function is possible in thiscase. However it emerged later that the new definition is a useful replacement forclassical one across all range of problems.
RLANGEN PROGRAM AT LARGE: OUTLINE 15
In the present note we will support the last claim by consideration of the sim-ple known problem: characterisation a n × n matrix up to similarity. Even that“freshman” question could be only sorted out by the classical spectral theory fora small set of diagonalisable matrices. Our solution in terms of new spectrum willbe full and thus unavoidably coincides with one given by the Jordan normal formof matrix. Other more difficult questions are the subject of ongoing research. Anything called “ func-tional calculus” uses properties of functions to model properties of operators . Thuschanging our viewpoint on functions, as was done in Section 3, we could get an-other approach to operators.The representation (3.4) is unitary irreducible when acts on the Hardy space H . Consequently we have one more reason to abolish the template definition 4.1: H is not an algebra. Instead we replace the homomorphism property by a symmetriccovariance : Definition 4.4. An analytic functional calculus for an element a ∈ A and an A -module M is a continuous linear mapping Φ : A ( D ) → A ( D , M ) such that(i) Φ is an intertwining operator Φρ = ρ a Φ between two representations of the SL ( R ) group ρ (3.4) and ρ a definedbelow in (4.4).(ii) There is an initialisation condition: Φ [ v ] = m for v ( z ) ≡ and m ∈ M ,where M is a left A -module.Note that our functional calculus released form the homomorphism conditioncan 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, wasthe Weyl functional calculus and was defined just by an integral formula as anoperator valued distribution [1]. In that paper (joint) spectrum was defined assupport of the Weyl calculus, i.e. as the set of point where this operator valueddistribution does not vanish. We also define the spectrum as a support of func-tional calculus, but due to our Definition 4.4 it will means the set of non-vanishingintertwining operators with primary subrepresentations. Definition 4.5.
A corresponding spectrum of a ∈ A is the support of the functionalcalculus Φ , i.e. the collection of intertwining operators of ρ a with prime representa-tions [10, § SL ( R ) in Banach Algebras. A simple but importantobservation is that the M ¨obius transformations (1.1) can be easily extended to anyBanach algebra. Let A be a Banach algebra with the unit e , an element a ∈ A with k a k < be fixed, then(4.2) g : a g · a = ( ¯ αa − ¯ βe )( αe − βa ) − , g ∈ SL ( R ) is a well defined SL ( R ) action on a subset A = { g · a | g ∈ SL ( R ) } ⊂ A , i.e. A is a SL ( R ) -homogeneous space. Let us define the resolvent function R ( g , a ) : A → A : R ( g , a ) = ( αe − βa ) − then(4.3) R ( g , a ) R ( g , g − a ) = R ( g g , a ) . The last identity is well known in representation theory [10, § induced representations . Thus we can again linearise (4.2) (cf. (3.2))in the space of continuous functions C ( A , M ) with values in a left A -module M ,e.g. M = A : ρ a ( g ) : f ( g − · a ) R ( g − g − , a ) f ( g − g − · a ) (4.4) = ( α ′ e − β ′ a ) − f (cid:18) ¯ α ′ · a − ¯ β ′ eα ′ e − β ′ a (cid:19) . For any m ∈ M we can again define a K -invariant vacuum vector as v m ( g − · a ) = m ⊗ v ( g − · a ) ∈ C ( A , M ) . It generates the associated with v m family of coherentstates v m ( u , a ) = ( ue − a ) − m , where u ∈ D .The wavelet transform defined by the same common formula based on coherentstates (cf. (3.3)): W m f ( g ) = h f , ρ a ( g ) v m i , is a version of Cauchy integral, which maps L ( A ) to C ( SL ( R ) , M ) . It is closely re-lated (but not identical!) to the Riesz-Dunford functional calculus: the traditionalfunctional calculus is given by the case: Φ : f W m f ( ) for M = A and m = e . The both conditions—the intertwining property and initial value—requiredby Definition 4.4 easily follows from our construction. ρ . Spectrum was defined in 4.5 as the support of our functional calculus. To elaborate its meaning we need the notionof a prolongation of representations introduced by S. Lie, see [34, 35] for a detailedexposition.
Definition 4.6. [35, Chap. 4] Two holomorphic functions have n th order contact ina point if their value and their first n derivatives agree at that point, in other wordstheir Taylor expansions are the same in first n + terms.A point ( z , u ( n ) ) = ( z , u , u , . . . , u n ) of the jet space J n ∼ D × C n is the equiv-alence class of holomorphic functions having n th contact at the point z with thepolynomial:(4.5) p n ( w ) = u n ( w − z ) n n ! + · · · + u ( w − z ) + u . For a fixed n each holomorphic function f : D → C has n th prolongation (or n -jet ) j n f : D → C n + :(4.6) j n f ( z ) = ( f ( z ) , f ′ ( z ) , . . . , f ( n ) ( z )) . The graph Γ ( n ) f of j n f is a submanifold of J n which is section of the jet bundle over D with a fibre C n + . We also introduce a notation J n for the map J n : f Γ ( n ) f ofa holomorphic f to the graph Γ ( n ) f of its n -jet j n f ( z ) (4.6).One can prolong any map of functions ψ : f ( z ) [ ψf ]( z ) to a map ψ ( n ) of n -jets by the formula(4.7) ψ ( n ) ( J n f ) = J n ( ψf ) . For example such a prolongation ρ ( n ) of the representation ρ of the group SL ( R ) in H ( D ) (as any other representation of a Lie group [35]) will be again a represen-tation of SL ( R ) . Equivalently we can say that J n intertwines ρ and ρ ( n ) : J n ρ ( g ) = ρ ( n ) ( g ) J n for all g ∈ SL ( R ) . RLANGEN PROGRAM AT LARGE: OUTLINE 17 (a)
XY λ λ λ λ (b) X Y λ λ λ λ Z (c) X Y λ λ λ λ Z F IGURE
8. Classical spectrum of the matrix from the Ex. 4.9 isshown at (a). Covariant spectrum of the same matrix in the jetspace is drawn at (b). The image of the covariant spectrum underthe map from Ex. 4.11 is presented (c).Of course, the representation ρ ( n ) is not irreducible: any jet subspace J k , k n is ρ ( n ) -invariant subspace of J n . However the representations ρ ( n ) are primary [10, § Proposition 4.7.
Let matrix a be a Jordan block of a length k with the eigenvalue λ = ,and m be its root vector of order k , i.e. a k − m = a k m = . Then the restriction of ρ a onthe subspace generated by v m is equivalent to the representation ρ k . Now we are pre-pared to describe a spectrum of a matrix. Since the functional calculus is an inter-twining operator its support is a decomposition into intertwining operators withprime representations (we could not expect generally that these prime subrepre-sentations are irreducible).Recall the transitive on D group of inner automorphisms of SL ( R ) , which cansend any λ ∈ D to and are actually parametrised by such a λ . This group extendsProposition 4.7 to the complete characterisation of ρ a for matrices. Proposition 4.8.
Representation ρ a is equivalent to a direct sum of the prolongations ρ ( k ) of ρ in the k th jet space J k intertwined with inner automorphisms. Consequentlythe spectrum of a (defined via the functional calculus Φ = W m ) labelled exactly by n pairs of numbers ( λ i , k i ) , λ i ∈ D , k i ∈ Z + , i n some of whom could coincide. Obviously this spectral theory is a fancy restatement of the
Jordan normal form of matrices.
Example 4.9.
Let J k ( λ ) denote the Jordan block of the length k for the eigenvalue λ . On the Fig. 8 there are two pictures of the spectrum for the matrix a = J ( λ ) ⊕ J ( λ ) ⊕ J ( λ ) ⊕ J ( λ ) , where λ = e iπ/ , λ = e i π/ , λ = e − i π/ , λ = e − iπ/ . Part (a) represents the conventional two-dimensional image of the spectrum, i.e.eigenvalues of a , and (b) describes spectrum sp a arising from the wavelet con-struction. The first image did not allow to distinguish a from many other essen-tially different matrices, e.g. the diagonal matrix diag ( λ , λ , λ , λ ) , which even have a different dimensionality. At the same time the Fig. 8(b) com-pletely characterise a up to a similarity. Note that each point of sp a on Fig. 8(b)corresponds to a particular root vector, which spans a primary subrepresentation. As was mentioned in the Introduction a res-onable spectrum should be linked to the corresponding functional calculus by anappropriate spectral mapping theorem. The new version of spectrum is based onprolongation of ρ into jet spaces (see Section 4.3). Naturally a correct version ofspectral 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 (4.7) we can define the pro-longation φ ( n ) ∗ onto the jet space J n . Its associated action ρ k φ ( n ) ∗ = φ ( n ) ∗ ρ n on thepairs ( λ , k ) is given by the formula:(4.8) φ ( n ) ∗ ( λ , k ) = (cid:18) φ ( λ ) , (cid:20) k deg λ φ (cid:21)(cid:19) , where deg λ φ 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 . Theorem 4.10 (Spectral mapping) . Let φ be a holomorphic mapping φ : D → D andits prolonged action φ ( n ) ∗ defined by (4.8) , then sp φ ( a ) = φ ( n ) ∗ sp a . The explicit expression of (4.8) for φ ( n ) ∗ , which involves derivatives of φ upto n th order, is known, see for example [7, Thm. 6.2.25], but was not recognised be-fore as form of spectral mapping. Example 4.11.
Let us continue with Example 4.9. Let φ map all four eigenvalues λ , . . . , λ of the matrix a into themselves. Then Fig. 8(a) will represent the classicalspectrum of φ ( a ) as well as a .However Fig. 8(c) shows mapping of the new spectrum for the case φ has orders of zeros at these points as follows: the order at λ , exactly the order at λ ,an order at least at λ , and finally any order at λ .
5. Open Problems
In this section we indicate several directions for further work which go throughthree main areas described in the paper..
Geometry is most elaborated area so far, yet many directionsare waiting for further exploration.(i) M ¨obius transformations (1.1) with three types of imaginary units ap-pear from the action of the group SL ( R ) on the homogeneous space SL ( R ) /H [28], where H is any subgroup A , N , K from the Iwasawa de-composition (1.3). Which other actions and hypercomplex numbers canbe obtained from semisimple Lie groups and their subgroups?(ii) Lobachevsky geometry of the upper half-plane is extremely beautifuland well-developed subject [4, 6]. However the traditional study is lim-ited to one subtype out of nine possible: with the complex numbers forM ¨obius transformation and the complex imaginary unit used in FSCc (2.2).The remaining eight cases shall be explored in various directions, no-tably in the context of discrete subgroups [3].(iii) The Filmore-Springer-Cnops construction, see subsection 2.2, is closelyrelated to the orbit method [11] applied to SL ( R ) . An extension of the RLANGEN PROGRAM AT LARGE: OUTLINE 19 orbit method from the Lie algebra dual to matrices representing cyclesmay be fruitful for semisimple Lie groups.
It is known that in several dimensions there are dif-ferent notions of analyticity, e.g. several complex variables and Clifford analysis.However, analytic functions of a complex variable are usually thought to be theonly options in a plane domain. The following seems to be promising:(i) Development of the basic components of analytic function theory (theCauchy integral, the Taylor expansion, the Cauchy-Riemann and Laplaceequations, etc.) from the same construction and principles in the elliptic,parabolic and hyperbolic cases and subcases.(ii) Identification of Hilbert spaces of analytic functions of Hardy and Bergmantypes, investigation of their properties. Consideration of the correspond-ing T ¨oplitz operators and algebras generated by them.(iii) Application of analytic methods to elliptic, parabolic and hyperbolicequations and corresponding boundary and initial values problems.(iv) Generalisation of the results obtained to higher dimensional spaces. De-tailed investigation of physically significant cases of three and four di-mensions.
The functional calculus of a finite dimensional op-erator considered in Section 4 is elementary but provides a coherent and compre-hensive treatment. It shall be extended to further cases where other approachesseems to be rather limited.(i) Nilpotent and quasinilpotent operators have the most trivial spectrumpossible (the single point { } ) while their structure can be highly non-trivial. Thus the standard spectrum is insufficient for this class of opera-tors. In contract, the covariant calculus and the spectrum give completedescription of nilpotent operators—the basic prototypes of quasinilpo-tent ones. For quasinilpotent operators the construction will be morecomplicated and shall use analytic functions mentioned in 5.2.i.(ii) The version of covariant calculus described above is based on the discreteseries representations of SL ( R ) group and is particularly suitable for thedescription of the discrete spectrum (note the remarkable coincidence inthe names).It is interesting to develop similar covariant calculi based on the twoother representation series of SL ( R ) : principal and complementary [32].The corresponding versions of analytic function theories for principal [16]and complementary series [24] were initiated within a unifying frame-work. The classification of analytic function theories into elliptic, para-bolic, hyperbolic [24, 27] hints the following associative chains: Representations of SL ( R ) — Function Theory — Type of Spectrum discrete series — elliptic — discrete spectrumprincipal series — hyperbolic —continuous spectrumcomplementary series — parabolic — residual spectrum(iii) Let a be an operator with sp a ∈ ¯ D and (cid:13)(cid:13) a k (cid:13)(cid:13) < Ck p . It is typical toconsider instead of a the power bounded operator ra , where < r < ,and consequently develop its H ∞ calculus. However such a regulari-sation is very rough and hides the nature of extreme points of sp a . To restore full information a subsequent limit transition r → of the regu-larisation parameter r is required. This make the entire technique rathercumbersome and many results have an indirect nature.The regularisation a k → a k /k p is more natural and accurate forpolynomially bounded operators. However it cannot be achieved withinthe homomorphic calculus Defn. 4.1 because it is not compatible withany algebra homomorphism. Albeit this may be achieved within thecovariant calculus Defn. 4.4 and Bergman type space from 5.2.ii.(iv) Several non-commuting operators are especially difficult to treat withfunctional calculus Defn. 4.1 or a joint spectrum. For example, deepinsights on joint spectrum of commuting tuples [36] refused to be gen-eralised to non-commuting case so far. The covariant calculus was ini-tiated [14] as a new approach to this hard problem and was later founduseful elsewhere as well. Multidimensional covariant calculus [21] shalluse analytic functions described in 5.2.iv. Due to the space restrictions we did not mentionedconnections with quantum mechanics [15, 22, 25, 26, 29, 30]. In general Erlangenapproach is much more popular among physicists rather than mathematicians.Nevertheless its potential is not exhausted even there.(i) There is a possibility to build representation of the Heisenberg groupusing characters of its centre with values in dual and double numbersrather than in complex ones. This will naturally unifies classical me-chanics, traditional QM and hyperbolic QM [9].(ii) Representations of nilpotent Lie groups with multidimensional centresin Clifford algebras as a framework for consistent quantum filed theoriesbased on De Donder–Weyl formalism [25].
Remark 5.1.
This work is performed within the “Erlangen programme at large”framework [24, 27], thus it would be suitable to explain the numbering of vari-ous papers. Since the logical order may be different from chronological one thefollowing numbering scheme is used:Prefix Branch description“0” or no prefix Mainly geometrical works, within the classical field of Er-langen programme by F. Klein, see [24, 28]“1” Papers on analytical functions theories and wavelets,e.g. [16]“2” Papers on operator theory, functional calculi and spectra,e.g. [23]“3” Papers on mathematical physics, e.g. [30]For example, this is the first paper in the mathematical physics area.
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