Subinvariants for Analytic Mappings on Matrix Balls
The Poincaré metric on the unit disk is invar iant under conformai m a p s of the disk o n t o itself and contrac ted by any analyt ic m a p of the disk in to itself. O n a d o m a i n G in several complex d imensions there arc many invariant metrics and some of these arc cont rac ted and some are not. We arc led by a na tura l quest ion in engineering to study G = .tfM„ — the unit ball o f » χ η matr ices and the b iho lomorph ic invariants produced by the 'cross rat io ' of two matrices (this includes all of the classical metrics which we know on ,SM„). 1. The paper characterizes all analytic maps of.if M „ into itself which contract the invariants produced by the cross ratio. In par t icular we prove that the linear f ract ional m a p s .#"(S) = (/1.S" + ß)(C.S + D)" 1 with matr ix coefficients A.B,C,D which m a p .iSM„ into itself have this proper ty . For example, they contract the Siegel metric. 2. We actually d o a m o r e general study involving the cross rat io of a matrix in with an arbi t rary matrix and prove tha t linear fract ional m a p s cont rac t the invariants in the more general s i tuat ion. Suppose one is given an amplifying device A and wants to use it as the central c o m p o n e n t in an amplifier. The objective is to design passive circuitry (or lossless circuitry) which when connected with A results in an amplifier with large gain. A na tura l quest ion is: can one build a better amplifier (more gain) with passive connect ing circuitry than with lossless connect ing circui try? This paper proves that the answer is no. In fact, this physical s ta tement is the precise content of 1. and 2. 1. proves it for a reflection type amplifier and 2. proves it for a (linearized) t ransis tor amplifier. AMS-Class i f icat ion: 32, 93, 47 This paper concerns the unit ball âêMn o í η χ η matrices with norm strictly less than 1 and analytic mappings of 3âM„ into itself. There are many metrics on ä#M„ which are invariant under biholomorphic mappings oí 3âM„ onto itself. The best known are the Caratheodory metric and the Siegel metric. The Caratheodory metric has the property that any analytic map f o í 3 S M n into itself contracts the Caratheodory distance between two elements of âSM„ (see [3]). For η = 1 the Siegel, Caratheodory and other invariant metrics are identical so they are all contracted by analytic maps of the disk into itself. It 218 J. A. Ball and J. W. Helton is natural to ask if all biholomorphically invariant metrics on ¿MMn are contracted fa b when n > 1. The answer is a resounding no. For example, the linear map 3F\ ^ Φ ^ of & M 2 into itself is easily seen to increase the Siegel distance between ^ I and \ | for small e. o o y V The purpose of this article is to analyze the issue of which maps ¡F are contractions in the Siegel and all related metrics. The example above is so simple that one might expect almost no J^'s to be contractive. We show that there actually is a very nice class of maps (GL(2n) acting on J M „ by fractional linear transformation) which are contractive. This result was motivated directly by electric circuit theory and has physical applications which are described in [1, §8]. We begin a more precise discussion by describing specifically the types of biholomorphic invariants we shall consider. They are all based on the cross ratio of two matrices S,Heá?M„ £(S, H) = (H — S) (S*H— 1) (S* — H*)(l —SH*)'. This has positive eigenvalues and is invariant up to similarity under biholomorphic maps of 3SM„, that is, δ (3(S), S ( i f ) ) = W~ i{S, H)W for some invertible nxn matrix W. Thus the eigenvalues V({S,H) for <?(S,/i) are basic invariants for biholomorphic mappings. Here<?=l , . . . ,n and • It is easy to make metrics from the ιy, for example, one basic class of metric is ρ (S, H) = arctanh Σ h(S,HY for q> 1. When q = 2 we get the Siegel metric; also lim gq(S,H) = v¡(S,H) is the Caratheodory metric. However, henceforth we shall just study the v¿(S,H) directly. We shall say that a mapping of á?M„ into itself is totally contractive if and only if ve(^{S),ß{H))<v^(S,H) for any S , / / e J M „ holds for 1 <i<n. This implies that both the Siegel and Caratheodory metric are contracted. Our first main result is an exact characterization of those analytic mappings which are totally contractive. To state our result precisely we give some notation. Let M be the set of all nxn matrices with rank <{ , and let y be the set of all analytic mappings ^:ââM„—>ââM„ such that for all Meâ#M„, + + (1.2) Subinvariants for Analytic Mappings on Matrix Balls 219 for 0<ί<η; that is, for any MeäSM„ and G in 3Λ. such that M + Ge<MM„, one has & (M + G)-&(M)<=;Ai The main result of § 1 is Theorem 1.1. An analytic map !F\36M„—>â$Mn is totally contractive if and only if $F is in i f . An important class of maps of MMn into itself is the linear fractional maps with matrix coefficients. A major consequence of Theorem 1.1 is that these maps are totally "α ß~ contractive. To be precise let M„ denote the « χ η matrices. Recall that if M = M2„, then the associated linear fractional map is given as SM(S) = (« S + ßHKS + y y ^ S e M J is in whenever the inverse exists. If M is in the group U(n,n) = {M\M*JM = J] when /„ 0 0 / „ jA , then maps the matrix ball onto itself. More generally, if M is only a J-contraction, that is, M belongs to the semigroup C(n, n) = {MeMn\M*JM <J}, then maps into itself. The heralded corollary is Corollary 1.2. The maps DMwith M in C(n, n) are totally contractive. This follows from Theorem 1.1 and the easily verified fact that each with M in C(n,n) is in i f . The corollary has implications for power transfer in circuits. When one connects a source with internal characteristic Seâ#M„ to a load with characteristic He0$M„, then the eigenvalues of S(S,H) are modes of the power mismatch between them. The corollary says that if one connects passive (energy dissipating) components to this circuit in a certain basic way, then all modes of the power mismatch are diminished. To study amplifiers one needs to generalize Corollary 1.2 to S e f M „ and HeMn but H$ââMn. That is the subject of § 2 of the paper and the second part of the introduction. Physically the eigenvalues here correspond to modes of the power gain of the amplifier. These modes are preserved when one inserts energy conserving circuitry and Corollary 1.4 says that the gain is always diminished with passive circuitry. A reader with any interest in this topic must see [1] for precise statements. Now we describe these generalizations. Most of the properties of S (S, H) mentioned above extend beyond matrices S,H in 3SM„. After all S"(S,H) is defined densely on M„ and the invariance (up to similarity) under the linear fractional maps &M(MeU(n,n)) continues to hold as long as all of the inverses in the formulas for ê, &M(S), and exist. The eigenvalues of S{S,H) will be positive provided either S or H lies in 0SM„. The singular values of a matrix M are the eigenvalues of |M|=(M*M)z; let M(f,j,k) denote the set of matrices with (singular values < 1 ,j singular values = 1 and k singular values > 1. We let ,/e be the class of all analytic mappings êP in if which have analytic continuations to a dense subset of Mn, and are such that whenever MeM(n — k, 0, k) is in the domain of SF, then (1.2) holds for all (> k. Finally, suppose J^ is an analytic mapping 220 J. A. Ball and J. W. Helton of ääM„ into itself with analytic continuat ion to a dense subset of M„. We say that J* is totally contractive on M„, if( 1.1) is satisfied for ( > kfor all Η in and SeM(n — k, 0, k) which are in the domain of . In § 2 we prove Theorem 1.3. An analytic mapp ing 3F of M„ into itself with analytic cont inuat ion to a dense subset of M„ is totally contract ive on Λ/„ if and only if is in i f e . Corollary 1.4. If MeC(n,n), then CM is totally contract ive on Mn. § 1. Analytic mappings on âSMn. It is well known (see e. g. [1] § 1) that the group .9= M\Mel)(n,n)} acts transitively on I M „ ; indeed, given Se38M„, if we set „ (R~ SQ~ \ *[S*R-> Q j where R*R = I SS* and Q*Q = I S*S, then Mse U(n, n) and S?M(.(0) = S. We use this to give a convenient equivalent definition of the semigroup ,f defined in the introduct ion. Lemma 1.1 .An analytic mapping •.3SM„-+36M„ is in if if and only if,for all Se34Mn the mapping J^· = P k f matrices into rank (matrices for 0<t <n, that is, Proof. Suppose .Φ$ maps into itself. Then, for where R 'e .^f , since any linear fract ional m a p such as is in f f (by the resolvent identity). Thus ,nS +R) = .?o 9 „ ^ p + R)) = JFoVXis(R') f o r s o m e "'<*t = <fw (0) + R"' f o r s o m e R'"eMe = &(S) + R"' and thus Conversely, suppose is in , ( f . Then since f f is a semigroup, is in f f for any S in 3SM„, and by definition J^.(0) = 0. It now follows simply f rom the definitions that J ^ maps into itself for 0 < i < n . Subinvariants for Analytic Mappings on Matrix Balls 221 Next note that ó'(S,0) = SS* for any SeM„. Thus the following is a special case of Theorem 1.1. Theorem 1.2. Let <8 be a transformation in ,f with i#(0) = 0 (such as above). Then for all Με.#Μ„, e.v. ¿(y (M Y$(M'f) < e.v.((MM*). (Here e.r.,(x)>. . .>e.v.„(x) are the eigenvalues of a matrix χ having nonnegative eigenvalues.) Proof. The proof is an adaptation of the usual proof of Schwarz's lemma. Let Me3SMn have singular value decomposition