Certain non-homogeneous matricial domains and Pick-Nevanlinna interpolation problem
aa r X i v : . [ m a t h . C V ] S e p CERTAIN NON-HOMOGENEOUS MATRICIAL DOMAINS ANDPICK–NEVANLINNA INTERPOLATION PROBLEM
VIKRAMJEET SINGH CHANDEL
Abstract.
In this article, we consider certain matricial domains that are naturally associ-ated 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 resultshows – generalizing a result of Ransford-White for the spectral unit ball – that the holo-morphic automorphism group of these matricial domains does not act transitively. We alsoconsider 2-point and 3-point Pick–Nevanlinna interpolation problem from the unit disc tothese matricial domains. We present results providing necessary conditions for the existenceof a holomorphic interpolant for these problems. In particular, we shall observe that theseresults are generalizations of the results provided by Bharali and Chandel related to theseproblems. Introduction and statement of results
Let D denote the open unit disc in the complex plane C centered at 0. Given a domainΩ ⊆ C and n ∈ N , n ≥
2, we define: S n (Ω) := { A ∈ M n ( C ) : σ ( A ) ⊂ Ω } . Here, M n ( C ) denotes the set of all n × n complex matrices, and σ ( A ) of a matrix A ∈ M n ( C )denotes the set of eigenvalues of A . It is not difficult to check that S n (Ω) is an open, connectedsubset of M n ( C ). Hence, by definition, considered as a subset of C n , it is a domain. In thecase Ω = D , the set S n (Ω) is known in the literature as the spectral unit ball and is denotedby Ω n .In the first part of this article, we shall establish that the holomorphic automorphismgroup of S n (Ω), denoted by Aut ( S n (Ω)), does not act transitively on S n (Ω) for most planardomains Ω of the complex plane. In particular, this result generalizes an analogous resultdue to Ransford-White (see [16]) about the spectral unit ball Ω n , n ≥
2. To state this resultprecisely, we need to introduce a few more objects associated naturally to S n (Ω).First, we consider the symmetrization map π n : C n −→ C n defined by: π n ( z ) := (cid:0) π n, ( z ) , . . . , π n, j ( z ) , . . . , π n, n ( z ) (cid:1) , where π n, j ( z ) is the j -th elementary symmetric polynomial. In other words, if we write π n, j ( z , . . . , z n ) = S j ( z , . . . , z n ), then S j satisfy: n Y j =1 ( t − z j ) = t n + n X j =1 ( − j S j ( z , . . . , z n ) t n − j , t ∈ C . The symmetrization map π n is a proper holomorphic map from C n to C n . Consider the n -thsymmetrized product of Ω, defined by:Σ n (Ω) := π n (Ω n ) . Mathematics Subject Classification.
Primary: 32H35, 30E05, 47A56; Secondary: 32F45, 47A60.
Key words and phrases. symmetrized product, invariant pseudo-distances, holomorphic automorphismsand proper maps, spectral unit ball, holomorphic functional calculus, Pick–Nevanlinna interpolation problem.Vikramjeet Singh Chandel is supported by an institute postdoctoral fellowship of IIT Bombay.
It is easy to see that Σ n (Ω) is a domain in C n .We now need to introduce another object before we state our first result. Given a holo-morphic self map φ of Ω, there is a holomorphic self map that φ induces on Σ n (Ω); namely:Σ n φ : Σ n (Ω) −→ Σ n (Ω) defined byΣ n φ (cid:0) π n ( z , . . . , z n ) (cid:1) := π n ( φ ( z ) , . . . , φ ( z n )) ∀ ( z , . . . , z n ) ∈ Ω n , where π n is the symmetrization map as defined above.We can now state our first main theorem as alluded to in the second paragraph above: Theorem 1.1.
Let Ω ⊂ C be a domain and let n ∈ N , n ≥ , be such that C \ Ω) ≥ n .Moreover, suppose the n -th symmetrized product of Ω , Σ n (Ω) , has the property: ( P ) for every proper holomorphic map Φ : Σ n (Ω) −→ Σ n (Ω) , there exists φ : Ω −→ Ω , aproper holomorphic map, such that Φ ≡ Σ n φ .Then for every holomorphic automorphism Ψ ∈ Aut ( S n (Ω)) there exists a holomorphic au-tomorphism ψ of Ω such that σ (Ψ( A )) = ψ ( σ ( A )) ∀ A ∈ S n (Ω) . In particular, the holomorphic automorphisms of S n (Ω) does not act transitively on S n (Ω) . We shall present a proof of the above theorem in Section 2. We first record the followingimportant remark regarding Theorem 1.1.
Remark . Let Ω ⊂ C be a bounded domain, then the condition on the cardinality of C \ Ωin the above theorem is trivially satisfied. It is a nontrivial result of Chakrabarty–Gorai [3,Corollary 1.3] — who generalized the analogous result of Edigarian–Zwonek [8, Theorem 1]for Σ n ( D ) — that Σ n (Ω) satisfies the property ( P ) above. The proof of this latter fact as givenin [3] crucially depended on the ability to extract subsequences – given an auxiliary sequenceconstructed from the given proper map – that converge locally uniformly.Given Ω, as in the statement of Theorem 1.1, Zwonek in [20, Theorem 16] proved that notonly Σ n (Ω) is Kobayashi hyperbolic (see Section 2 for the definition), in fact, it is Kobayashicomplete. Now, since Kobayashi complete domains are taut, the same proof as in [3] could befollowed to establish that Σ n (Ω) has the property ( P ) above. So, in fact, under the conditionon the domain Ω as in Theorem 1.1, Σ n (Ω) satisfies the condition ( P ) automatically.As mentioned before, Theorem 1.1 provides a generalization of part ( a ) of Theorem 4 in[16] due to Ransford–White, for a class of domains Ω which satisfy the conditions stated inthe statement of Theorem 1.1.One could also ask what happens if the map Ψ in the statement of Theorem 1.1 is aproper holomorphic map instead of a holomorphic automorphism? We address this questiontoo here, and we show that under certain restrictions on the domain Ω; namely: Ω is ahyperconvex domain, such that Σ n (Ω) satisfies the condition ( P ) above then an analogousresult similar to the conclusion of Theorem 1.1 holds true.Before we state this result, we recall that a domain D ⊂ C n is called hyperconvex if thereexists a negative plurisubharmonic exhaustion function on D . We shall see in Section 2 that aif a domain Ω ⊂ C is hyperconvex then the cardinality of C \ Ω cannot be finite. In particular,hyperconvex domains satisfy the condition on the cardinality of C \ Ω as in the statement ofTheorem 1.1.Now we state our second result related to the proper holomorphic self-maps of S n (Ω). Theorem 1.3.
Let Ω ⊂ C be a hyperconvex domain and let n ∈ N , n ≥ , be given. Supposethe n -th symmetrized product of Ω , Σ n (Ω) , satisfies the condition ( P ) above. Then for every ON-HOMOGENEOUS MATRICIAL DOMAINS 3 proper holomorphic map
Ψ : S n (Ω) −→ S n (Ω) , there exists a proper holomorphic self-map ψ of Ω such that σ (Ψ( A )) = ψ ( σ ( A )) ∀ A ∈ S n (Ω) . We shall present a proof of Theorem 1.3 in Section 2. We state here some important obser-vations relevant to Theorem 1.3.
Remark . As mentioned before, a hyperconvex domain Ω must satisfy the condition on thecardinalilty of C \ Ω as in the statement of Theorem 1.1. Hence as noted in the Remark 1.2,it must also satisfy the condition ( P ) above automatically. Notice also that for hyperconvexdomains, Theorem 1.1 follows easily from that of Theorem 1.3. Although the significance ofTheorem 1.1 lies in that it holds true for a large class of domains containing hyperconvexdomains. Remark . Let Ω ⊂ C be a domain such that C \ Ω has finitely many connected componentsnone of which is a single point. Then it is proved in [6] that such domains are uniformsqueezing . Uniform squeezing domains are also called holomorphic homogeneous regulardomains. It is a fact due to Yeung that uniform squeezing domains are hyperconvex [19].Hence, such domains Ω in C for which C \ Ω has only finitely many connected componentsnone of which is single point, is a class of hyperconvex domains.
Remark . Let Ω be a domain as in Remark 1.5 and let p ∈ N denote the number ofconnected components of C \ Ω. Mueller-Rudin showed in [14] that when p ≥ p = 2, it is a fact that Ω isbiholomorphic to an annulus A r := { z ∈ C : r < | z | < } for some r, < r <
1. In the caseof annulus it is also known (see [14] for a reference) that every proper holomorphic self mapis a holomorphic automorphism.The second part of this article is devoted to the following Pick-Nevanlinna interpolationtype problem. • Given { ( ζ j , W j ) ∈ D × S n (Ω) : 1 ≤ j ≤ N } , N ≥ ζ j ’s being distinct, find neces-sary and sufficient conditions for the existence of a holomorphic map F ∈ O ( D , S n (Ω))such that F ( ζ j ) = W j for all j = 1 , . . . , N .In the case when such a function F exists, we shall say that F is an interpolant of the data { ( ζ j , W j ) ∈ D × S n (Ω) : 1 ≤ j ≤ N } .In this article, we shall only consider the above problem when N = 2 or N = 3. Startingwith N = 2, we shall provide a necessary condition for the existence of a holomorphicinterpolant. This necessary condition will remind the reader of the classical Schwarz lemmain one complex variables. But before we state this result, we need to introduce the followingpseudo-distance.The Carath´eodory pseudo-distance , denoted by C Ω , on a domain Ω in C is defined by: C Ω ( p, q ) := sup {M D ( f ( p ) , f ( q )) : f ∈ O (Ω , D ) } . (1.1)Here and elsewhere in this article M D ( z , z ) is the M¨obius distance between z and z ,defined as: M D ( z , z ) := (cid:12)(cid:12)(cid:12)(cid:12) z − z − z z (cid:12)(cid:12)(cid:12)(cid:12) ∀ z , z ∈ D . The reader will notice that we have defined C Ω in terms of the M¨obius distance rather thanthe hyperbolic distance on D . This is done purposely because most conclusions in metricgeometry that rely on C Ω are essentially unchanged if M D is replaced by the hyperbolic VIKRAMJEET SINGH CHANDEL distance on D in (1.1), and because the M¨obius distance arises naturally in the proofs of ourtheorems.A domain Ω ⊂ C will be called Carath´eodory hyperbolic if C Ω is a distance in the senseof metric spaces. It is easy to see that Ω is Carath´eodory hyperbolic if and only if H ∞ (Ω),the set of all bounded holomorphic functions in Ω, separates points in Ω; e.g. every boundeddomain is Carath´eodory hyperbolic. In what follows, we shall always consider domains Ωthat are Carath´eodory hyperbolic.We now present our first result concerning the interpolation problem above when N = 2. Theorem 1.7.
Let F ∈ O ( D , S n (Ω)) , n ≥ , and let ζ , ζ ∈ D . Write W j = F ( ζ j ) , and if λ ∈ σ ( W j ) , then let m ( λ ) denote the multiplicity of λ as a zero of the minimal polynomial of W j . Then: max max µ ∈ σ ( W ) Y λ ∈ σ ( W ) C Ω ( µ, λ ) m ( λ ) , max λ ∈ σ ( W ) Y µ ∈ σ ( W ) C Ω ( λ, µ ) m ( µ ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ζ − ζ − ζ ζ (cid:12)(cid:12)(cid:12)(cid:12) . (1.2)We shall present our proof of Theorem 1.7 in Section 4. When Ω = D , we know that C Ω ( z , z ) = M D ( z , z ). Substituting this into the inequality (1.2) establishes Theorem 1.5in [2] which is an important result related to the 2-point interpolation problem from D toΩ n . Hence Theorem 1.7 gives a generalization of Theorem 1.5 in [2] for all matricial domains S n (Ω).It is shown in [2] that when Ω = D the above result gives a necessary condition for theexistence of an interpolant for the 2-point interpolation problem that is inequivalent to thenecessary condition that are known in the literature [5, 15]. Moreover, in this case (Ω = D )when n ≥
3, there exists a 2-point data set for which (1.2) implies that the data cannot admitan interpolant whereas the condition in [5, 15] are inconclusive.At this point we wish to discuss an important tool that plays a crucial role in establishingthe inequality (1.2) and is at the heart of our next theorem related to the 3-point interpolationproblem. We begin with the extremal problem associated to the Carath´eodory pseudo-distance C Ω on a domain Ω in C . Recall: C Ω ( p, q ) := sup {M D ( f ( p ) , f ( q )) : f ∈ O (Ω , D ) } = sup {| f ( q ) | : f ∈ O (Ω , D ) : f ( p ) = 0 } . (1.3)The equality in (1.3) is due to the fact that the automorphism group of D acts transitivelyon D and the M¨obius distance is invariant under its action. Applying Montel’s Theorem, it iseasy to see that there exists a function g ∈ O (Ω , D ) such that g ( p ) = 0 and g ( q ) = C Ω ( p, q ).Such a function is called an extremal solution for the extremal problem determined by (1.3).It is a fact that for Carath´eodory hyperbolic domains there is a unique extremal solution (seethe last two paragraphs in [10]). Let us denote by G Ω ( p, q ; · ) the unique extremal solutiondetermined by the extremal problem (1.3). Definition 1.8.
Given A ∈ S n (Ω) and z ∈ Ω \ σ ( A ), consider the function: B ( A, z ; · ) := Y λ ∈ σ ( A ) G Ω ( λ, z ; · ) m ( λ ) (1.4)where G Ω ( λ, z ; · ) is the unique extremal solution corresponding to the pair ( λ, z ) as discussedabove and m ( λ ) is the multiplicity of λ ∈ σ ( A ) as a zero of the minimal polynomial of A .Observe that for every z ∈ Ω \ σ ( A ), B ( A, z ; · ) ∈ O (Ω , D ) and has a zero at each λ ∈ σ ( A )of multiplicity at least m ( λ ). For each z ∈ Ω \ σ ( A ), B ( A, z ; · ) induces, via the holomorphicfunctional calculus (which we will discuss in Section 3), a holomorphic map from S n (Ω) to Ω n ON-HOMOGENEOUS MATRICIAL DOMAINS 5 that maps A to 0 ∈ M n ( C ). This simple trick turns out to be quite important in addressingthe 3-point interpolation problem.Now we are ready to present the result of this article related to the 3-point interpolationproblem from D to S n (Ω). In what follows, B j, z will denote the function as defined in(1.4) — as well as its extension to S n (Ω) — associated to the matrix W j , j = 1 , , Theorem 1.9.
Let ζ , ζ , ζ ∈ D be distinct points and let W , W , W ∈ S n (Ω) , n ≥ . Let m ( j, λ ) denote the multiplicity of λ as a zero of the minimal polynomial of W j , j ∈ { , , } .Given j, k ∈ { , , } such that j = k , z ∈ Ω \ σ ( W k ) and ν ∈ D , we write: q z ( ν, j, k ) := max ((cid:20) m ( j, λ ) − ord λ B ′ k, z + 1 (cid:21) + 1 : λ ∈ σ ( W j ) ∩ B − k, z { ν } ) . Finally, for each k ∈ { , , } let G ( k ) := max ( { , , } \ { k } ) , and L ( k ) := min ( { , , } \ { k } ) . If there exists a map F ∈ O ( D , S n (Ω)) such that F ( ζ j ) = W j , j ∈ { , , } , then for each k ∈ { , , } , and z ∈ Ω \ σ ( W k ) we have: • either σ (cid:0) B k, z ( W G ( k ) ) (cid:1) ⊂ D (cid:0) , | ψ k ( ζ G ( k ) ) | (cid:1) , σ (cid:0) B k, z ( W L ( k ) ) (cid:1) ⊂ D (cid:0) , | ψ k ( ζ L ( k ) ) | (cid:1) and max max µ ∈ σ ( B k, z ( W L ( k ) )) Y ν ∈ σ ( B k, z ( W G ( k ) )) M D (cid:18) µψ k ( ζ L ( k ) ) , νψ k ( ζ G ( k ) ) (cid:19) q z ( ν, G ( k ) , k ) , max µ ∈ σ ( B k, z ( W G ( k ) )) Y ν ∈ σ ( B k, z ( W L ( k ) )) M D (cid:18) µψ k ( ζ G ( k ) ) , νψ k ( ζ L ( k ) ) (cid:19) q z ( ν, L ( k ) , k ) ≤ M D (cid:0) ζ L ( k ) , ζ G ( k ) (cid:1) • or there exists a θ z ∈ R such that B − k, z { e iθ z ψ k ( ζ G ( k ) ) } ⊆ σ ( W G ( k ) ) and B − k, z { e iθ z ψ k ( ζ L ( k ) ) } ⊆ σ ( W L ( k ) ) . Here, ψ j denotes the automorphism ψ j ( ζ ) := ( ζ − ζ j ) / (1 − ζ j ζ ) − , ζ ∈ D , of D and [ · ]denotes the greatest-integer function. Given a ∈ C and a function g that is holomorphic in aneighbourhood of a , ord a g will denote the order of vanishing of g at a (with the understandingthat ord a g = 0 if g does not vanish at a ). Remark . We shall present our proof of Theorem 1.9 in Section 6. The proof is stronglymotivated from the proof of Theorem 1.4 in [4]. In fact, Theorem 1.9 above is a generalizationof Theorem 1.4 in [4]. This is shown in Observation 6.1 by explicitly computing the function B k, z ( · ). It turns out that when Ω = D the statement of Theorem 1.9 coincides with thestatement of Theorem 1.4 in [4]. We also refer the reader to Remark 1.5 in [4] for thediscussion of how Theorem 1.4 in [4] is different from the other results that are present inthe literature related to the 3-point interpolation problem.2. Preliminaries and proofs of Theorem 1.1 and Theorem 1.3
In this section, we shall present our proofs of Theorem 1.1 and Theorem 1.3. At theheart of our proofs is a result that itself is quite interesting. It is Lemma 2.2 below. Butbefore we prove this lemma, we shall take a digression and recall the definition of Kobayashipseudo-distance and Kobayashi hyperbolicity.
VIKRAMJEET SINGH CHANDEL
The Kobayashi pseudo-distance.
Let D ⊂ C n be a domain and let h denote thehyperbolic distance on D . The Kobayashi pseudo-distance K D : D × D −→ [0 , ∞ ) is definedby: given two points p, q ∈ D , K D ( p, q ) := inf n n X i =1 h ( ζ i − , ζ i ) : ( φ , . . . , φ n ; ζ , . . . , ζ n ) ∈ A ( p, q ) o where A ( p, q ) is the set of all analytic chains in D joining p to q . Here, ( φ , . . . , φ n ; ζ , . . . , ζ n )is an analytic chain in D joining p to q if φ i ∈ O ( D , D ) for each i such that p = φ ( ζ ) , φ n ( ζ n ) = q and φ i ( ζ i ) = φ i +1 ( ζ i )for i = 1 , . . . , n − K D is a pseudo-distance. Using the Schwarz lemma inone complex variable one could see that K D ≡ h . One of the most important properties thatthe Kobayashi pseudo-distance enjoys is the following: if F : D −→ D is a holomorphicmap, then K D (cid:0) F ( p ) , F ( q ) (cid:1) ≤ K D ( p, q ) for all p, q ∈ D .A domain D ⊂ C n is called Kobayashi hyperbolic if the pseudo-distance K D is a truedistance, i.e., K D ( p, q ) = 0 if and only if p = q . The collection of all bounded domains isan example of Kobayashi hyperbolic domains. This follows from the observation that for abounded domain any co-ordinate function maps the domain into a disc, which is Kobayashihyperbolic. This together with contracting property of K D under the holomorphic mapsestablishes that bounded domains are hyperbolic. We refer the interested reader to [12,Chapter 3] for a comprehensive account on Kobayashi pseudo-distance.It is a fact that K C d ≡ d ≥
1. This is not difficult to prove but we skip the proof ofthis fact here (see [12, Chapter 3]). The following is a generalization of Liouville’s Theoremand is obvious:
Result 2.1.
Let D ⊂ C n be a Kobayashi hyperbolic domain. Let F : C d −→ D be aholomorphic map then F is a constant function. We need one more tool to state the lemma alluded to at the beginning of this sec-tion. Given a matrix A ∈ M n ( C ), we write its characteristic polynomial as χ ( A )( t ) = t n + P nk =1 ( − k χ k ( A ) t n − k , where χ k ( A ) are polynomials in the entries of A . It is obviousthat this naturally leads to a map χ : S n (Ω) −→ Σ n (Ω) defined by: χ ( A ) = (cid:0) χ ( A ) , . . . , χ n ( A ) (cid:1) , where χ k ( A )’s are as above. Notice that χ is a holomorphic map from S n (Ω) to Σ n (Ω).Now we can state the lemma which implies that a holomorphic self map of S n (Ω) preservesthe spectra of matrices. More precisely: Lemma 2.2.
Let Ω ⊂ C be a domain and let n ∈ N , n ≥ , be such that C \ Ω) ≥ n .Let F : S n (Ω) −→ S n (Ω) be a holomorphic self map. Then for every A, B ∈ S n (Ω) such that χ ( A ) = χ ( B ) , we have χ ( F ( A )) = χ ( F ( B )) .Proof. Let D be a diagonal matrix such that χ ( D ) = χ ( A ). We know that there exists C ∈ M n ( C ) and an strictly upper triangualr matrix U such that A = exp( − C ) ( D + U ) exp( C ) . Now consider the map f : C −→ M n ( C ) defined by f ( ζ ) := exp( − C ζ ) ( D + ζ U ) exp( Cζ ) ∀ ζ ∈ C . Notice that χ ( f ( ζ )) = χ ( D + ζ U ) = χ ( D ), hence f ( C ) ⊂ S n (Ω). This lets us to define themap Ψ( ζ ) := χ ◦ F ◦ f ( ζ ) for all ζ ∈ C . It is obvious that Ψ is a holomorphic map from C toΣ n (Ω). ON-HOMOGENEOUS MATRICIAL DOMAINS 7
Under the condition on Ω as in the statement of the lemma, a result of Zwonek [20,Theorem 16] implies that Σ n (Ω) is Kobayashi hyperbolic. Then Result 2.1 implies that Ψ isa constant function. Hence, Ψ(0) = χ ( F ( D )) = Ψ(1) = χ ( F ( A )). Proceeding similarly weget χ ( F ( D )) = χ ( F ( B )). This establishes the lemma. (cid:3) The proof of the above lemma is motivated from that of Theorem 1 in [16] by Ransford–White. But it is a far reaching generalization of Theorem 1 in [16]; e.g., when Ω is anyCarath´eodory hyperbolic domain, then C \ Ω) cannot be finite. In particular, Carath´eodoryhyperbolic domains satisfy the condition as in the statement of Lemma 2.2.We shall now present our proof of Theorem 1.1.2.2.
The proof of Theorem 1.1.
Proof.
Consider a relation G from Σ n (Ω) into Σ n (Ω) defined by: G ( X ) := χ ◦ Ψ ◦ χ − ( { X } ) ∀ X ∈ Σ n (Ω) . (2.1)Here Ψ is as in the statement of Theorem 1.1. From Lemma 2.2, it follows that for each X ∈ Σ n (Ω), G ( X ) is a singleton set. Hence G : Σ n (Ω) −→ Σ n (Ω) is a well defined map. Claim. G is holomorphic.To see this, choose an arbitrary X ∈ Σ n (Ω) and fix it. Now consider the polynomial P X [ t ] := t n + P nj =1 ( − j X j t n − j . We define a map τ : Σ n (Ω) −→ M n ( C ) by setting: τ ( X ) := C (cid:0) P X (cid:1) (2.2)where C (cid:0) P X (cid:1) denotes the companion matrix of the polynomial P X . Recall: given a monicpolynomial of degree k of the form p [ t ] = t k + P kj =1 a j t k − j , where a j ∈ C , the companionmatrix of p is the matrix C ( p ) ∈ M k ( C ) given by C ( p ) := − a k − a k − . . . . . . ... − a k × k . It is a fact that χ ( C ( p ))( t ) = p ( t ). From this, it follows that τ is holomorphic and χ ◦ τ = I on Σ n (Ω). This, in particular, implies that τ ( X ) ∈ χ − { X } . Applying Lemma 2.2 again, wesee that χ ◦ Ψ ◦ χ − ( { X } ) = χ ◦ Ψ ◦ τ ( X ), i.e., G ( X ) = χ ◦ Ψ ◦ τ ( X ). Since each of the maps χ, Ψ , τ are holomorphic, the claim follows. Claim. G ∈ Aut (cid:0) Σ n (Ω) (cid:1) .To see this, consider H : Σ n (Ω) −→ Σ n (Ω) defined by H ( X ) := χ ◦ Ψ − ◦ χ − ( { X } ) for all X ∈ Σ n (Ω). Exactly the same argument as above shows that H ( X ) = χ ◦ Ψ − ◦ τ ( X ). Fromthis and that χ ◦ τ = I on Σ n (Ω), we get that G ◦ H = I , H ◦ G = I . This establishes that G as defined in (2.1) is a holomorphic automorphism of Σ n (Ω).Since Σ n (Ω) satisfies property ( P ), in the statement of Theorem 1.1, it is not difficult tosee that there exists ψ , a holomorphic automorphism of Ω, such that: G ◦ π n ( z , . . . , z n ) := π n (cid:0) ψ ( z ) , . . . , ψ ( z n ) (cid:1) for all ( z , . . . , z n ) ∈ Ω n .Now let A ∈ S n (Ω) be given and suppose ( λ , . . . , λ n ) and ( µ , . . . , µ n ) is a list of eigen-values of A and Ψ( A ) respectively, repeated according to their multiplicity as a zeros ofcharacteristic polynomial. Then from the definition of G and χ , we have G ◦ π n ( λ , . . . , λ n ) = π n ( µ , . . . , µ n ). On the other hand from the equation above we get G ◦ π n ( λ , . . . , λ n ) = VIKRAMJEET SINGH CHANDEL π n ( ψ ( λ ) , . . . , ψ ( λ n )). This implies π n ( ψ ( λ ) , . . . , ψ ( λ n )) = π n ( µ , . . . , µ n ). This in particu-lar implies that σ (Ψ( A )) = ψ ( σ ( A )). Since A is arbitrary, this establishes the conclusion ofour theorem. (cid:3) A few more Preliminaries.
In this subsection, we shall gather a few more tools thatare crucial to our proof of Theorem 1.3. We shall present our proof of Theorem 1.3 in thenext subsection.Recall the construction of the function f in the proof of Lemma 2.2. In particular, it showsthat for any A ∈ S n (Ω) there exists an entire function f into S n (Ω) such that f (1) = A and f (0) = D , where D is a diagonal matrix such that χ ( D ) = χ ( A ). Using this property, weshall prove a proposition regarding the pluri-complex Green function for the domain S n (Ω).Before we do this, let us first recall the pluri-complex Green function for a domain in C n .Let D ⊂ C n be a domain and let P := { ( p j , m j ) ∈ D × R + : 1 ≤ j ≤ N } be a set of poleswith p j = p k when j = k . Following Lelong [13], we define the pluri-complex Green functionwith poles in P by: g D ( P ; w ) := sup (cid:8) ν ( w ) : ν ∈ P SH (cid:0) D, [ −∞ , (cid:1) and such that ν ( z ) − m j log || z − p j || is bounded from above in a neighborhood of p j , j = 1 , . . . , N (cid:9) Here,
P SH (cid:0) D, [ −∞ , (cid:1) denotes the set of all negative pluri-subharmonic functions on D . Incase m j = 1 for all j , we shall write g D ( p , . . . , p N ; · ) in place of g D ( P ; · ). Now we can stateand prove the proposition alluded to in the previous paragraph regarding the pluri-complexGreen function for S n (Ω). Proposition 2.3.
Let
A, B ∈ S n (Ω) . Then for any D , a diagonal matrix with χ ( D ) = χ ( B ) we have: g S n (Ω) ( A ; B ) = g S n (Ω) ( A ; D ) . Proof.
As discussed above, let f : C −→ S n (Ω) be a holomorphic map such that f (0) = D and f (1) = B . Consider now u : C −→ [ −∞ ,
0) defined by u ( ζ ) := g S n (Ω) ( A ; f ( ζ )). Then u is a bounded subharmonic function defined on C . Hence u has to be constant. In particular, g S n (Ω) ( A ; B ) = g S n (Ω) ( A ; f (1)) = g S n (Ω) ( A ; f (0)) = g S n (Ω) ( A ; D ). (cid:3) Let Ω ⊂ C be a hyperconvex domain. By definition, there is a negative subharmonicexhaustion function u on Ω. In particular, u is a non-constant, bounded above, subharmonicfunction on Ω. This implies if we let E = C \ Ω then E = ∅ . The set E cannot be a polarset (see [17, Section 3.2] for the definition of polar set) otherwise by Theorem 3.6.7 in [17], itwill follow that u is a constant function, which will be a contradiction. Now it is a fact thatfinite and countable subsets of C are polar (see [17, Section 3.2]), from this it follows that E is uncountable.From the above discussion we see that given a hyperconvex domain Ω ⊂ C , and a positiveinteger n, n ≥
2, C \ Ω) ≥ n . Hence, Σ n (Ω) is Kobayashi hyperbolic for every hyperconvexdomain Ω. We also wish to state that Zwonek in the article [20, Proposition 11] proved thatΣ n (Ω) is hyperconvex if and only if Ω is hyperconvex. Putting all this together what we haveis that if Ω is hyperconvex then for each n ∈ N , n ≥
2, the domains Σ n (Ω) are Kobayashihyperbolic and hyperconvex.Now we are in a position to present our proof of Theorem 1.3.2.4. The proof of Theorem 1.3.
Proof.
We consider the function G : Σ n (Ω) −→ Σ n (Ω) defined by G ( X ) = χ ◦ Ψ ◦ χ − { X } .From the discussion above, we know that Ω satisfies the hypothesis as in Lemma 2.2. Hence ON-HOMOGENEOUS MATRICIAL DOMAINS 9 by Lemma 2.2, G is well defined. Exactly proceeding as in the proof of Theorem 1.1, we alsosee that G is holomorphic. Claim. G is a proper holomorphic self map of Σ n (Ω).To establish this it will be sufficient to prove that if { X ν } ⊂ Σ n (Ω) is a sequence, having nolimit points in Σ n (Ω), then { G ( X ν ) } has no limit points in Σ n (Ω). Assume, on the contrary,a sequence { X ν } ⊂ Σ n (Ω), having no limit points in Σ n (Ω), such that { G ( X ν ) } has a limitpoint in Σ n (Ω). This implies that there is a subsequence of { G ( X ν ) } , that we continue todenote with { G ( X ν ) } , such that { G ( X ν ) } converges to a point X ∈ Σ n (Ω).Recall the map τ : Σ n (Ω) −→ S n (Ω) as in the proof of Theorem 1.1. Let { λ ,ν , . . . , λ n,ν } be a list of eigenvalues of Ψ( τ ( X ν )) repeated according to their multiplicity as a zero of thecharacteristic polynomial of Ψ( τ ( X ν )). Then, using the property that χ ◦ τ ≡ I , we have: G ( X ν ) = χ (cid:0) Ψ( τ ( X ν )) (cid:1) = χ (cid:0) diag[ λ ,ν , . . . , λ n,ν ] (cid:1) = π n (cid:0) λ ,ν , . . . , λ n,ν (cid:1) (2.3)where diag[ λ ,ν , . . . , λ n,ν ] denotes the diagonal matrix with entries λ j,ν . Now using the proper-ness of π n (cid:12)(cid:12) Ω n : Ω n −→ Σ n (Ω) – and that { G ( X ν ) } converges to X ∈ Σ n (Ω) – there exists asubsequence of (cid:8) Λ ν ∈ Ω n : Λ ν = ( λ ,ν , . . . , λ n,ν ) (cid:9) , which we continue to denote by { Λ ν } ,such that { Λ ν } converges to Λ = ( λ , , . . . , λ n, ) ∈ Ω n . This, owing to equation (2.3) above,implies: π n (cid:0) λ , , . . . , λ n, (cid:1) = χ (cid:0) diag[ λ , , . . . , λ n, ] (cid:1) = X . Let A ∈ S n (Ω) be such that A is not a critical value of Ψ. Let N be the multiplicity of Ψand suppose (cid:8) B , . . . , B N (cid:9) = Ψ − { A } . The upper semicontinuity of g S n (Ω) implies g S n (Ω) (cid:0) A ; diag[ λ , , . . . , λ n, ] (cid:1) ≥ lim ν →∞ g S n (Ω) (cid:0) A ; diag[ λ ,ν , . . . , λ n,ν ] (cid:1) . (2.4)Now by Proposition 2.3, and the behaviour of Green function under proper holomorphicmappings (see [9, Theorem 1.2]) we get: g S n (Ω) (cid:0) A ; diag[ λ ,ν , . . . , λ n,ν ] (cid:1) = g S n (Ω) (cid:0) A ; Ψ( τ ( X ν )) (cid:1) = g S n (Ω) (cid:0) B , . . . , B N ; τ ( X ν ) (cid:1) ≥ N X j =1 g S n (Ω) (cid:0) B j ; τ ( X ν ) (cid:1) ≥ N X j =1 g Σ n (Ω) (cid:0) χ ( B j ) ; X ν (cid:1) . (2.5)Since Σ n (Ω) is hyperconvex and { X ν } is a sequence that does not have a limit point inΣ n (Ω), we have lim ν →∞ g Σ n (Ω) (cid:0) χ ( B j ) ; X ν (cid:1) → j, ≤ j ≤ N ; see e.g. [13]. Fromthis, inequality (2.5) and (2.4) it follows that g S n (Ω) (cid:0) A ; diag[ λ , , . . . , λ n, ] (cid:1) = 0, which isa contradiction. Hence our assumption that { G ( X ν ) } has a subsequence that converges inΣ n (Ω) is false. This establishes that G is a proper holomorphic self map of Σ n (Ω).Since Σ n (Ω) has the property ( P ), there exists ψ , a proper holomorphic self map of Ω, suchthat G = Σ n ψ . Using this and proceeding similarly as in the last paragraph in the proof ofTheorem 1.1, we get the desired result. (cid:3) A brief survey of holomorphic functional calculus
A very essential part of our proofs of Theorem 1.7 and Theorem 1.9 below is the ability,given a domain Ω ⊂ C and a matrix A ∈ S n (Ω), to define f ( A ) in a meaningful way foreach f ∈ O (Ω). Most readers will be aware that this is what is known as the holomorphicfunctional calculus. We briefly recapitulate the holomorphic functional calculus and its basicproperties in a setting which will be relevant to our proofs in the coming sections.Throughout this section, X will denote a finite dimensional complex Banach space and T a linear operator in B ( X ) := the set of all bounded linear operators on X . The symbol I will denote the identity operator and we will interpret T = I . In what follows, we shall denoteby C [ t ] the set of all polynomials with complex coefficients in the indeterminate t . Given apolynomial P ∈ C [ t ], if we write P ( t ) := P ni =0 α i t i with α i ∈ C , then by P ( T ) we will meanthe sum P ni =0 α i T i .For a fix T ∈ B ( X ) and λ ∈ C , we consider the set (cid:8) ( λ I − T ) j : j ∈ N (cid:9) . If λ / ∈ σ ( T ), wenotice that Ker( λ I − T ) j = { } for each j ∈ N . On the other hand if λ ∈ σ ( T ) and if wedefine V jλ := Ker( λ I − T ) j then we have { } ⊆ V λ ⊆ V λ ⊆ · · · ⊆ V jλ ⊆ . . . . Since X is finitedimensional, there is a k , k ≤ dim( X ), such that V kλ = V k +1 λ = V jλ for all j ≥ k + 1. Definition 3.1.
Let T ∈ B ( X ) and let λ ∈ C . Then the index of λ , m ( λ ), is defined by: m ( λ ) := min (cid:8) j ∈ N : Ker( λ I − T ) j = Ker( λ I − T ) j +1 (cid:9) . Notice that m ( λ ) = 0 if and only if λ / ∈ σ ( T ). A question arises at this point: givenpolynomials P, Q ∈ C [ t ], and T ∈ B ( X ), when P ( T ) = Q ( T )? We state a very importantresult that among other things primarily provides answer to this question: Result 3.2. [7, Chapter 7, Section 1]
Let
P, Q ∈ C [ t ] , and let T ∈ B ( X ) . Then we have P ( T ) = Q ( T ) if and only if each λ ∈ σ ( T ) is a zero of P − Q of order m ( λ ) . We refer the reader to [7, Chapter 7, Section 1] for a proof of this result. The proof as givenin [7] also shows that m ( λ ) is the multiplicity of λ as a zero of the minimal polynomial of T .Given T ∈ B ( X ), let F ( T ) be the set of all holomorphic functions in a neighbourhood of σ ( T ). Given f ∈ F ( T ), we define: f ( T ) := P ( T ) , where P ∈ C [ t ] with f ( j ) ( λ ) = P ( j ) ( λ ), 0 ≤ j ≤ m ( λ ) − ∀ λ ∈ σ ( T ) . Here, f ( j ) , P ( j ) denotes the j -th derivative of f and P respectively. It follows from Result 3.2that the definition above is unambiguous and the function Θ T : F ( T ) −→ B ( X ) defined byΘ T ( f ) := f ( T ) has following properties: if f, g ∈ F ( T ) and α, β ∈ C then: • αf + βg ∈ F ( T ), and Θ T ( αf + βg ) = α Θ T ( f ) + β Θ T ( g ), • f g ∈ F ( T ) and Θ T ( f g ) = Θ T ( f ) Θ T ( g ), • σ (Θ T ( f )) = σ ( f ( T )) = f ( σ ( T )).The last property above is called the Spectral Mapping Property in literature. We also notethat from the second property above, it follows that f ( T ) g ( T ) = g ( T ) f ( T ) for all f, g ∈ F ( T ).Given p ∈ C , let e p ( · ) be a function that is identically equal to 1 in a neighbourhoodof p , and identically equal to 0 in a neighbourhood of each point of σ ( T ) ∩ ( C \ { p } ). Set E ( p ) = e p ( T ). Then we have following: Result 3.3. [7, Chapter 7, Section 1]
Given T ∈ B ( X ) and p ∈ C , let E ( p ) ∈ B ( X ) be asdefined above then we have: • E ( p ) = 0 if and only if p ∈ σ ( T ) . • E ( p ) = E ( p ) and E ( p ) E ( q ) = 0 for p = q . • I = P p ∈ σ ( T ) E ( p ) .Remark . If { λ , . . . , λ k } be an enumeration of σ ( T ), and let X i = E ( λ i ) X . Result 3.3implies that X = X ⊕ · · · ⊕ X k . Moreover, since
T E ( λ i ) = E ( λ i ) T , it follows that T X i ⊆ X i , i = 1 , . . . , k . Thus, to thedecomposition of the spectrum σ ( T ) into k points there corresponds a direct sum decompo-sition of X into k invariant subspaces of T . Thus the study of the action of T on X may bereduced to the study of the action of T on each of the subspaces X i . ON-HOMOGENEOUS MATRICIAL DOMAINS 11
Remark . If we restrict T on X i and write T = λ i I + ( T − λ i I ) on X i . Then since theholomorphic function ( t − λ i ) m ( λ i ) e λ i ( t ) has a zero of order m ( λ i ) at each point of σ ( T ), wehave ( T − λ i I ) m ( λ i ) E ( λ i ) = 0. Clearly ( T − λ i I ) m ( λ i ) − E ( λ i ) = 0. This shows that restrictedto X i , the operator T − λ i I is a nilpotent operator of order m ( λ i ). Thus, in each space X i ,the operator T is the sum of a scalar multiple λ i I of the identity and a nilpotent operator T − λ i I of order m ( λ i ).We now state a result that gives an explicit formula for computing f ( T ) in terms of theprojections E ( λ ), λ ∈ σ ( T ). Result 3.6.
Let T ∈ B ( X ) and let f ∈ F ( T ) be given. Then f ( T ) = X λ ∈ σ ( T ) m ( λ ) − X j =0 ( T − λ I ) j j ! f ( j ) ( λ ) E ( λ ) . (3.1)The formula (3.1) follows immediately from the properties of the function Θ T described above;once we observe the following: given f ∈ F ( T ), consider the function g ∈ F ( T ) defined by: g ( t ) := X λ ∈ σ ( T ) m ( λ ) − X j =0 ( t − λ ) j j ! f ( j ) ( λ ) e λ ( t ) , then we have f ( i ) ( λ ) = g ( i ) ( λ ) , i ≤ m ( λ ) −
1, for λ ∈ σ ( T ).4. The proof of Theorem 1.7
We shall present our proof of Theorem 1.7 in this section. Before we present our proof, weshall need certain complex analytic tools related to the spectral unit ball. We first discussthem in the next subsection.4.1.
Complex analytic properties of the spectral unit ball.
For n ∈ Z + , recall thespectral unit ball, Ω n ⊂ C n , is the collection of all matrices A ∈ M n ( C ) whose spectrum σ ( A ) is contained in D . We also recall the definition of spectral radius ρ of a matrix A definedby ρ ( A ) := max (cid:8) | λ | : λ ∈ σ ( A ) (cid:9) . We first state the result: Result 4.1 (Janardhanan, [11]) . The spectral unit ball, Ω n , is an unbounded, balanced,pseudo-convex domain with Minkowski function given by the spectral radius ρ . It is a fact that the Minkowski function of a balanced pseudo-convex domain is pluri-subharmonic (see [12, Appendix B.7.6]). Hence, it follows from this result that ρ | Ω n ispluri-subharmonic. The pluri-subharmonicity of spectral radius function also follows fromanother important result due to Vesentini [18]. This latter result regarding the spectralradius function is for a general Banach algebra.We now state an important lemma for holomorphic functions in O ( D , Ω n ). This lemmacould be considered as a generalization of the Schwarz lemma for holomorphic functions in O ( D , D ). Lemma 4.2.
Let F ∈ O ( D , Ω n ) be such that F (0) = 0 . Then there exists G ∈ O ( D , Ω n ) such that F ( ζ ) = ζ G ( ζ ) for all ζ ∈ D . In particular, we have ρ ( F ( ζ )) ≤ | ζ | for all ζ ∈ D . The lemma is a consequence of the fact that ρ | Ω n is pluri-subharmonic. We do not wish topresent a proof of the above lemma here; we refer the interested reader to [4, Lemma 4.3]for a proof. We also wish to state another another lemma which is a consequence of thepluri-subharmonicity of the spectral radius function. Lemma 4.3.
Let Φ ∈ O ( D , Ω n ) be such that there exists a θ ∈ R and ζ ∈ D satisfying e iθ ∈ σ (Φ( ζ )) . Then e iθ ∈ σ (Φ( ζ )) for all ζ ∈ D . Lemma 4.3 is not needed in the proof of Theorem 1.7 although it will be an important toolin the proof of Theorem 1.9. We stated it here since it follows from the pluri-subharmonicityof spectral radius function; see e.g. [4, Section 4]. We are now ready to present our proof ofTheorem 1.7.4.2.
The proof of Theorem 1.7.
Proof.
Let F ∈ O ( D , S n (Ω)) be such that F ( ζ j ) = W j , j = 1 ,
2. For each k ∈ { , } , considerΦ k ∈ O ( D , S n (Ω)) defined by: Φ k ( ζ ) = F ◦ ψ − k ( ζ ) ∀ ζ ∈ D . (4.1)Here, ψ k is the automorphism ψ k ( ζ ) := ( ζ − ζ k )(1 − ζ k ζ ) − , ζ ∈ D , of D . Then Φ k (0) = W k and Φ k ( ψ k ( ζ j )) = W j , j = k . Now for an arbitrary but fixed z ∈ Ω \ σ ( W k ), consider B k, z ∈ O (Ω , D ) defined by: B k, z ( · ) := Y λ ∈ σ ( W k ) G Ω ( λ, z ; · ) m ( λ ) . (4.2)Observe that B k, z ( · ) = B ( W k , z ; · ) is as in Definition 1.8.As B k, z ∈ O (Ω), it induces — via the holomorphic functional calculus — a map (which wecontinue to denote by B k, z ) from S n (Ω) to M n ( C ). The Spectral Mapping Theorem tells usthat σ ( B k, z ( X )) = B k, z ( σ ( X )) ⊂ D for every X ∈ S n (Ω). Hence B k, z ( X ) ⊂ Ω n for every X ∈ S n (Ω). Claim. B k, z (Φ k (0)) = 0.To see this we write: B k, z ( ζ ) = (cid:16) Y λ ∈ σ ( W k ) ( ζ − λ ) m ( λ ) (cid:17) g z ( ζ )for some g z ∈ O (Ω) and for every ζ ∈ Ω. Hence, since — by the holomorphic functional calcu-lus — the assignment f f (Φ k (0)) , f ∈ O (Ω), is multiplicative, as discussed in Section 3,we get B k, z (Φ k (0)) = (cid:16) Y λ ∈ σ ( W k ) (Φ k (0) − λ I ) m ( λ ) (cid:17) g z (Φ k (0)) . Now since the minimal polynomial for W k = Φ k (0) is given by Q λ ∈ σ ( W k ) ( t − λ ) m ( λ ) , we seethat the product term in the right hand side of the above equation is zero, whence the claim.Consider the map Ψ k, z defined by:Ψ k, z ( ζ ) := B k, z ◦ Φ k ( ζ ) , ζ ∈ D . It is a fact that Ψ k, z ∈ O ( D ). Moreover, from the above claim and the discussion just beforeit, we have Ψ k, z ∈ O ( D , Ω n ) with Ψ k, z (0) = 0. By Lemma 4.2, we get that ρ (Ψ k, z ( ζ )) ≤ | ζ | ∀ ζ ∈ D . Now from the definition of Ψ k, z and from the Spectral Mapping Theorem we get σ (Ψ k, z ( ζ )) = σ ( B k, z (Φ k ( ζ ))) = B k, z ( σ (Φ k ( ζ ))). This together with the above equation gives us: | B k, z ( µ ) | ≤ | ζ | ∀ ζ ∈ D and µ ∈ σ (Φ k ( ζ )) . We put ζ = ψ k ( ζ j ) in the above equation. Then, since Φ k ( ψ k ( ζ j )) = W j , by the aboveequation and (4.2), we get: | B k, z ( µ ) | = Y λ ∈ σ ( W k ) | G Ω ( λ, z ; µ ) | m ( λ ) ≤ | ψ k ( ζ j ) | ∀ µ ∈ σ ( W j ) . (4.3) ON-HOMOGENEOUS MATRICIAL DOMAINS 13
Since z is arbitrary, for an arbitrary but fixed µ ∈ σ ( W j ), we can take z = µ in the aboveequation. This with the observation that G Ω ( λ, µ ; µ ) = C Ω ( λ, µ ) gives us that: Y λ ∈ σ ( W k ) C Ω ( λ, µ ) m ( λ ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ζ − ζ − ζ ζ (cid:12)(cid:12)(cid:12)(cid:12) ∀ µ ∈ σ ( W j ) . (4.4)Interchanging the roles of j and k , in the above discussion will give an inequality of the form(4.4) with j and k interchanged. The inequality (1.2) will follow from these two inequalities. (cid:3) Minimal polynomials under holomorphic functional calculus
In this section, we develop the key matricial tool needed in establishing Theorem 1.9, whichis the computation of the minimal polynomial for f ( A ), given f ∈ O (Ω) and A ∈ S n (Ω), n ≥
2. This is the content of Theorem 5.2 below. In what follows, given integers p < q ,[ p . . q ] will denote the set of integers { p, p + 1 , . . . , q } . Given A ∈ M n ( C ), we will denote itsminimal polynomial by M A . We begin with a lemma: Lemma 5.1 (Chandel, [4]) . Let ( α , α , . . . , α n − ) ∈ C n , n ≥ . Let A = P n − j =0 α j N j ,where N is the nilpotent operator of degree n . Then the minimal polynomial for A is givenby: M A ( t ) = ( t − α ) [( n − /l ( α ,α ,...,α n − )]+1 . (5.1)Here, [ · ] denotes the greatest integer function and l ( α , α , . . . , α n − ) is defined by: l ( α , α , . . . , α n − ) := ( n, if α j = 0 ∀ j ∈ [1 . . n − , min { j ∈ [1 . . n −
1] : α j = 0 } , otherwise . The reader is referred to [4, Lemma 3.1] for a proof of this lemma.Given a ∈ C and g a holomorphic function in a neighbourhood of a , ord a g will denote theorder of vanishing of g at a , as defined after the statement of Theorem 1.9. We now presentthe main result of this section. Theorem 5.2.
Let A ∈ S n (Ω) , n ≥ , and let f ∈ O (Ω) be a non-constant function. Supposethat the minimal polynomial for A is given by M A ( t ) = Y λ ∈ σ ( A ) ( t − λ ) m ( λ ) . Then the minimal polynomial for f ( A ) is given by M f ( A ) ( t ) = Y ν ∈ f ( σ ( A )) ( t − ν ) k ( ν ) , where, k ( ν ) = max (cid:26)(cid:20) m ( λ ) − ord λ f ′ + 1 (cid:21) + 1 : λ ∈ σ ( A ) ∩ f − { ν } (cid:27) . Proof.
Let λ ∈ σ ( A ) and let E ( λ ) be the projection operator as defined in Section 3 with T = A . Then by Remark 3.4 we know that: X = ⊕ λ ∈ σ ( A ) X λ , where X λ = E ( λ ) X , λ ∈ σ ( A ). (5.2)Note that X = C n , but it really does not matter here. Let x ∈ X and write x = ⊕ λ ∈ σ ( A ) x λ ,where x λ = E ( λ ) x . Then by Result 3.6 we have: f ( A ) x = X λ ∈ σ ( A ) m ( λ ) − X j =0 ( A − λ I ) j j ! f ( j ) ( λ ) E ( λ ) ! x = X λ ∈ σ ( A ) m ( λ ) − X j =0 ( A − λ I ) j j ! f ( j ) ( λ ) ! x λ . Now consider the operators A λ : X λ −→ X defined by: A λ := m ( λ ) − X j =0 ( A − λ I ) j j ! f ( j ) ( λ ) . Since A X λ ⊆ X λ for all λ ∈ σ ( A ), A λ leaves X λ invariant; i.e. A λ ( X λ ) ⊆ X λ . Hence f ( A ) x = ⊕ λ ∈ σ ( A ) A λ x λ , where x = ⊕ λ ∈ σ ( A ) x λ is as in (5.2). (5.3)This establishes that f ( A ) = ⊕ λ ∈ σ ( A ) A λ .Focussing our attention to A λ , we first observe that – by Remark 3.5 – the operator A − λ I , λ ∈ σ ( A ), restricted to X λ is a nilpotent operator of degree m ( λ ) −
1. Hence by Lemma 5.1,we have M A λ ( t ) := ( t − f ( λ )) ν , where ν = (cid:20) m ( λ ) − l ( f ′ ( λ ) , f ′′ ( λ ) , . . . , f ( m ( λ ) − ( λ )) (cid:21) + 1 . If ord λ f ′ ≤ m ( λ ) −
2, then l ( f ′ ( λ ) , f ′′ ( λ ) , . . . , f ( m ( λ ) − ( λ )) = ord λ f ′ + 1, else ord λ f ′ + 1 > ( m ( λ ) −
1) and l ( f ′ ( λ ) , f ′′ ( λ ) , . . . , f ( m ( λ ) − ( λ )) > ( m ( λ ) − (cid:20) m ( λ ) − l ( f ′ ( λ ) , f ′′ ( λ ) , . . . , f ( m ( λ ) − ( λ )) (cid:21) = (cid:20) m ( λ ) − ord λ f ′ + 1 (cid:21) . From the last two expressions M A λ ( t ) = ( t − f ( λ )) (cid:20) m ( λ ) − ord λ f ′ + 1 (cid:21) . Let us rewrite (5.3) in another way: f ( A ) = ⊕ ν ∈ f ( σ ( A )) B ν , where B ν = ⊕ f ( λ )= ν A λ The minimal polynomial for f ( A ), M f ( A ) , is the least common multiple of M B ν , ν ∈ f ( σ ( A )).Notice that the polynomials M B ν , ν ∈ f ( σ ( A )) are relatively prime to each other. Hence M f ( A ) ( t ) = Y ν ∈ f ( σ ( A )) M B ν ( t )Now the minimal polynomial for B ν is the least common multiple of minimal polynomials of A λ such that f ( λ ) = ν , λ ∈ σ ( A ). It is easy to see (using the expression for M A λ ) that M B ν ( t ) = ( t − ν ) k ( ν ) , where k ( ν ) = max ((cid:20) m ( λ ) − ord λ f ′ + 1 (cid:21) : λ ∈ f − { ν } ∩ σ ( A ) ) From the last two expressions, we get the desired result. (cid:3)
Remark . Theorem 5.2 is a generalization of Theorem 3.4 in [4] which dealt with the caseΩ = D . The proof of Theorem 3.4 as presented in [4] is very specific to the unit disc. This isbecause the proof of Lemma 3.2 – which is a crucial tool in establishing Theorem 3.4 in [4] –exploits a property of a holomorphic function in the unit disc; namely: every holomorphicfunction in D has a power series representation on D . This latter property is very specific tothe unit disc. ON-HOMOGENEOUS MATRICIAL DOMAINS 15 The proof of Theorem 1.9
In this section, we shall present our proof of Theorem 1.9. After the proof, we shall alsopresent an observation that, among other things, shows that when Ω = D , Theorem 1.4 in [4]is the same as our theorem, in particular, [4, Theorem 1.4] is a special case of Theorem 1.9.Before we present our proof, we wish to restate the inequality (1.2) when Ω = D , so thatit is easy to apply in our proof of Theorem 1.9. First of all, as mentioned before, in this case C Ω ( z , z ) = M D ( z , z ). Now given W j ∈ Ω n , j = 1 ,
2, consider: b j ( t ) = Y λ ∈ σ ( W j ) (cid:18) t − λ − λt (cid:19) m ( j, λ ) . Here m ( j, λ ) denotes the multiplicity of λ as a zero of the minimal polynomial for W j .The finite Blaschke product b j is called the minimal Blaschke product corresponding to W j , j = 1 ,
2. With this notation in hand, we can restate the inequality (1.2) as:max (cid:26) max µ ∈ σ ( W ) | b ( µ ) | , max λ ∈ σ ( W ) | b ( λ ) | (cid:27) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ζ − ζ − ζ ζ (cid:12)(cid:12)(cid:12)(cid:12) . (6.1)It is in this form that we shall use the inequality (1.2) in our proof when Ω = D . We nowpresent our proof of Theorem 1.9.6.1. The proof of Theorem 1.9.
Let F ∈ O ( D , S n (Ω)) be such that F ( ζ j ) = W j , j ∈{ , , } . Let k ∈ { , , } and let us denote by B k, z ( · ) the function B ( W k , z ; · ), where z ∈ Ω \ σ ( W k ) be any arbitrary but fixed point in Ω. Now consider the function: e F k, z := B k, z ◦ F ◦ ψ − k . Here ψ k ( ζ ) = ( ζ − ζ k ) / (1 − ¯ ζ k ζ ) is an automorphism of the unit disc that maps ζ k to 0. Then e F k, z ∈ O ( D , Ω n ) such that e F k, z ( ψ k ( ζ L ( k ) )) = B k, z ( W L ( k ) ) , e F k, z ( ψ k ( ζ G ( k ) )) = B k, z ( W G ( k ) )and e F k, z (0) = 0. By Lemma 4.2, we get e F k, z ( ζ ) = ζ e G k, z ( ζ ) ∀ ζ ∈ D , for some e G k, z ∈ O ( D , Ω n ) . (6.2)Two cases arise: Case 1. e G k, z ( D ) ⊂ Ω n .In view of (6.2), we have e G k, z (cid:0) ψ k ( ζ L ( k ) ) (cid:1) = W L ( k ) , k, z and e G k, z (cid:0) ψ k ( ζ G ( k ) ) (cid:1) = W G ( k ) , k, z (6.3)where W L ( k ) , k, z := B k, z ( W L ( k ) ) (cid:14) ψ k ( ζ L ( k ) ) and W G ( k ) , k, z := B k, z ( W G ( k ) ) (cid:14) ψ k ( ζ G ( k ) ). Nowusing the inequality (6.1), a necessary condition for (6.3) ismax ( max η ∈ σ ( W L ( k ) , k, z ) | b G ( k ) , k, z ( η ) | , max η ∈ σ ( W G ( k ) , k, z ) | b L ( k ) , k, z ( η ) | ) ≤ M D (cid:0) ζ G ( k ) , ζ L ( k ) (cid:1) , (6.4)where b L ( k ) , k, z and b G ( k ) , k, z denote the minimal Blaschke product corresponding to the ma-trices W L ( k ) , k, z , W G ( k ) , k, z . Given the definitions of the latter matrices, we will need Theo-rem 5.2 to determine b L ( k ) , k, z , b G ( k ) , k, z . By this theorem, we have b L ( k ) , k, z ( t ) = Y ν ∈ σ ( B k, z ( W L ( k ) ) ) (cid:18) t − ν/ψ k ( ζ L ( k ) )1 − ν/ψ k ( ζ L ( k ) ) t (cid:19) q z ( ν, L ( k ) , k ) (6.5) b G ( k ) , k, z ( t ) = Y ν ∈ σ ( B k, z ( W G ( k ) ) ) (cid:18) t − ν/ψ k ( ζ G ( k ) )1 − ν/ψ k ( ζ G ( k ) ) t (cid:19) q z ( ν, G ( k ) , k ) , (6.6) where q z ( ν, L ( k ) , k ) and q z ( ν, G ( k ) , k ) are as in the statement of Theorem 1.9. Now if η ∈ σ ( W L ( k ) , k, z ) or η ∈ σ ( W G ( k ) , k, z ), then η = µ/ψ k ( ζ L ( k ) ) for some µ ∈ σ ( B k, z ( W L ( k ) )) or η = µ/ψ k ( ζ G ( k ) ) for some µ ∈ σ ( B k, z ( W G ( k ) )), respectively, and conversely. This observationtogether with (6.6), (6.5) and (6.4) establishes the first part of our theorem. Case 2. e G k, z ( D ) ∩ ∂ Ω n = ∅ .Let ζ ∈ D be such that e iθ z ∈ σ ( e G k, z ( ζ )) for some θ z ∈ R . By Lemma 4.3, we have e iθ z ∈ σ ( e G k, z ( ζ )) for every ζ ∈ D . By (6.2), e iθ z ζ ∈ σ ( e F k, z ( ζ )). Let Φ ≡ F ◦ ψ − k . ThenΦ ∈ O (cid:0) D , S n (Ω) (cid:1) and we have: e iθ z ζ ∈ σ (cid:0) B k, z ◦ Φ( ζ ) (cid:1) = B k, z { σ (Φ( ζ )) } ∀ ζ ∈ D , where the last equality is an application of the Spectral Mapping Theorem. For each ζ ∈ D ,let ω ζ ∈ σ (Φ( ζ )) be such that B k, z ( ω ζ ) = e iθ z ζ . Notice that if ζ = ζ then ω ζ = ω ζ ,whence E := { ω ζ : ζ ∈ D } is an uncountable set in Ω. Notice that ω ζ satisfies: B k, z ( ω ζ ) = e iθ z ζ and det ( ω ζ I − Φ( ζ )) = 0 ∀ ζ ∈ D . As E is uncountable, it follows from the identity principle that the holomorphic map x det (cid:0) x I − Φ( e − iθ z B k, z ( x )) (cid:1) is identically 0 on Ω. As B k, z maps Ω into D , it followsthat B − k, z { e iθ z ζ } ⊂ σ (Φ( ζ )) = σ (cid:0) F ◦ ψ − k ( ζ ) (cid:1) ∀ ζ ∈ D . (6.7)Putting ζ = ψ k ( ζ L ( k ) ) and ψ k ( ζ G ( k ) ) respectively in (6.7) we get B − k, z { e iθ z ψ k ( ζ L ( k ) ) } ⊂ σ ( F ( ζ L ( k ) )) = σ ( W L ( k ) ) and B − k, z { e iθ z ψ k ( ζ G ( k ) ) } ⊂ σ ( F ( ζ G ( k ) )) = σ ( W G ( k ) ). (cid:3) Observation 6.1.
When
Ω = D , it is not difficult to show that G D ( λ, z ; ζ ) := V ( λ, z ) ζ − λ − ¯ λζ where V ( λ, z ) ∈ T such that V ( λ, z ) (cid:0) ( z − λ ) / (1 − ¯ λz ) (cid:1) = M D ( λ, z ) . Hence when Ω = D , and A ∈ Ω n then B ( A, z ; · ) is of the form B ( A, z ; · ) := R (cid:0) σ ( A ) , z (cid:1) Y λ ∈ σ ( A ) (cid:18) ζ − λ − ¯ λζ (cid:19) m ( λ ) . Here, R (cid:0) σ ( A ) , z (cid:1) is a uni-modular constant. In particular, B ( A, z ; · ) at any point z is ascalar multiple of the minimal Blaschke product corresponding to A by a uni-modular constantdepending on z .In this case, given W i ∈ Ω n , i ∈ { , , } as in the statement of Theorem 1.9, we noticethat: ord λ B ′ k, z = ord λ B ′ k , where B k ( · ) = B W k ( · ) is the minimal Blaschke product corresponding to W k . Hence thenumbers q z ( ν, j, k ) does not depend on z . Also, since M D ( z , z ) = M D ( e iθ z , e iθ z ) , wehave: max µ ∈ σ ( B k, z ( W L ( k ) ) ) Y ν ∈ σ ( B k, z ( W G ( k ) ) ) M D (cid:18) µψ k ( ζ L ( k ) ) , νψ k ( ζ G ( k ) ) (cid:19) q ( ν, G ( k ) , k, z ) = max µ ∈ σ ( B k ( W L ( k ) ) ) Y ν ∈ σ ( B k ( W G ( k ) ) ) M D (cid:18) µψ k ( ζ L ( k ) ) , νψ k ( ζ G ( k ) ) (cid:19) q ( ν, G ( k ) , k ) . The above equality is also true when we replace G ( k ) by L ( k ) and vice-versa. It follows fromthis that when Ω = D , the statement of Theorem 1.9 reduces to that of Theorem 1.4 in [4] . ON-HOMOGENEOUS MATRICIAL DOMAINS 17
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