Operator Monotone Functions and Löwner Functions of Several Variables - PDF

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Operator Monotone Functions and Löwner Functions of Several Variables Jim Agler John E. McCarthy N. J. Young June 30, 2013 Abstract We prove generalizations of Löwner s results on matrix monotone functions

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Operator Monotone Functions and Löwner Functions of Several Variables Jim Agler John E. McCarthy N. J. Young June 30, 2013 Abstract We prove generalizations of Löwner s results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna s theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone. 1 Introduction In 1934, K. Löwner published a very influential paper [25] studying functions on an open interval E R that are matrix monotone, i.e. functions f with the property that whenever S and T are self-adjoint matrices whose spectra are in E then S T f(s) f(t ). (1.1) This property is equivalent (see Subsection 1.3) to being locally matrix monotone, i.e. if S(t) is a C 1 arc of self-adjoint matrices with σ(s(t)) E then S (t) 0 d f(s(t)) 0. (1.2) dt Partially supported by National Science Foundation Grant DMS Partially supported by National Science Foundation Grant DMS Partially supported by London Mathematical Society Grant Roughly speaking, Löwner showed that if one fixes a dimension n and wants (1.1) or (1.2) to hold for n-by-n self-adjoint matrices, then certain matrices derived from the values of f must all be positive semidefinite. As n increases, the conditions become more restrictive. In the limit as n (equivalently, if one passes to self-adjoint operators on an infinite dimensional Hilbert space), then a necessary and sufficient condition is that the function f must have an analytic continuation to a function F that maps the upper half-plane to itself. The goal of this paper is to extend the above notions to several variables. In particular, we want to study functions of d variables applied to d-tuples of commuting self-adjoint operators. Given two d-tuples S = (S 1,..., S d ) and T = (T 1,..., T d ), we shall say that S T if and only if S r T r for every 1 r d. We want to study functions that satisfy (1.1) or (1.2) for d-tuples. Before we can describe our results, we must first give a more detailed description of the one-dimensional case. We recommend the book [13] by W. Donoghue for a well-written account from a modern perspective. See also the paper [30]. Note that there is another approach to extending Löwner s results to several variables where the operators S 1,..., S d act on different spaces H 1,..., H d, and f(s) is interpreted to act on H 1 H d. We refer the reader to the papers [16, 35, 23] and references therein. 1.1 Dimension one Let E be an open set in R, and let n 2 be a natural number. The Löwner class L 1 n(e) is the set of C 1 functions f : E R with the property that, whenever {x 1,..., x n } is a set of n distinct points in E, then the matrix A, defined by A ij = f(x j ) f(x i ) x j x i if i j f (x i ) if i = j, is positive semi-definite. We shall let M n denote the n-by-n complex matrices, SAM n the self-adjoint n-by-n matrices, and SA the bounded self-adjoint operators on an infinite dimensional separable Hilbert space. Definition 1.3. A function f is locally n-matrix monotone on the open set E R if, whenever S is in SAM n with σ(s) consisting of n 2 distinct points in E, and S(t) is a C 1 curve in SAM n with S(0) = S and d dt S(t) t=0 0, then d dt f(s(t)) t=0 0. Remark 1.4. This definition is slightly different from the one in the first paragraph, where the eigenvalues were not required to be distinct. We use this definition to be consistent with the multivariable Definition 1.9 below. However, using formula (6.6.31) in [18] for d dt f(s(t)), it is easy to show that in the one variable case the two different definitions are equivalent 1. We shall say that f is n-matrix monotone on E, or M n -monotone, if, whenever S and T are in SAM n and all their eigenvalues lie in E, then (1.1) holds. To emphasize the difference from locally monotone, we shall also call n-matrix monotone functions globally M n -monotone. Replacing SAM n by SA, we get the definitions of locally operator monotone and operator monotone. Theorem 1.5 (Löwner). Let E R be open, and let f C 1 (E). Then f is locally n-matrix monotone on E if and only if f is in L 1 n(e). We shall use Π to denote the upper half-plane, {z C : Im z 0}. Definition 1.6. Let E R be open. The Pick class on E, denoted P(E), is the set of real-valued functions f on E for which there exists an analytic function F : Π Π such that F extends analytically across E and 2 lim F (x + iy) = f(x) x E. y 0 Theorem 1.7 (Löwner). Let E R be open, and let f C 1 (E). The following are equivalent: (i) The function f is locally operator monotone on E. (ii) The function f is in L 1 n(e) for all n. (iii) The function f is in P(E). 1 This formula says that for a C 1 arc S(t) = U(t)Λ(t)U (t), with U(t) unitary and Λ(t) diagonal with diagonal entries λ 1 (t),..., λ n (t), and a C 1 function f, then d dt f(s(t)) = U(t) ([ f(λ i(t), λ j (t))] [U(t) S (t)u(t)]) U(t), where means the matrix of divided differences, and denotes the Schur product. 2 The notation y 0 means y decreases to 0. The notation r 1 means r increases to 1. 3 1.2 Dimension d 2 : Local results We shall let CSAM d n denote the set of d-tuples of commuting selfadjoint n-by-n matrices, and CSA d be the set of d-tuples of commuting self-adjoint bounded operators. If S is a commuting d-tuple of selfadjoint operators acting on the Hilbert space H, and f is a real-valued continuous (indeed, measurable) function on the joint spectrum of S in R d, then f(s) is a well-defined self-adjoint operator on H. Definition 1.8. Let E be an open set in R d, and f be a real-valued C 1 function on E. Say f is locally operator monotone on E if, whenever S is in CSA d with σ(s) E, and S(t) is a C 1 curve in CSA d with S(0) = S and d dt S(t) t=0 0, then d dt f(s(t)) t=0 exists and is 0. We shall not concern ourselves in this paper on what conditions on f guarantee that f(s(t)) is differentiable; for these see e.g. [28]. Definition 1.9. Let E be an open set in R d, and f be a real-valued C 1 function on E. We say f is locally M n -monotone on E if, whenever S is in CSAMn d with σ(s) = {x 1,..., x n } consisting of n distinct points in E, and S(t) is a C 1 curve in CSAMn d with S(0) = S and d dt S(t) t=0 0, then d dt f(s(t)) t=0 exists and is 0. We define the Löwner classes in d variables, L d n(e), by: Definition Let E be an open subset of R d. The set L d n(e) consists of all real-valued C 1 -functions on E that have the following property: whenever {x 1,..., x n } are n distinct points in E, there exist positive semi-definite n-by-n matrices A 1,..., A d so that A r f (i, i) = x r and f(x j ) f(x i ) = xi (x r j x r i )A r (i, j) 1 i, j n. Here is our d-variable version of Theorem 1.5; we prove it as Theorem Theorem Let E be an open set in R d, and f a real-valued C 1 function on E. Then f is locally M n -monotone if and only if f is in L d n(e). 4 In generalizations of Theorem 1.7, there turns out to be a difference between the case d = 2 and d 2. Definition The Löwner class, L d, is the set of functions F : Π d Π with the property that there exist d positive semi-definite kernel functions A r, 1 r d, on Π d such that F (z) F (w) = (z 1 w 1 )A 1 (z, w) (z d w d )A d (z, w). When d = 1 or 2, the Löwner class coincides with the set of all analytic functions from Π d to Π, but for d 3 it is a proper subset (see Remark 2.18). Definition Let E R d be open. The class L(E) is the set of real-valued functions f on E for which there exists an analytic function F in L d such that F extends analytically across E and lim F y 0 (x1 + iy,..., x d + iy) = f(x 1,..., x d ) x E. We prove the following result as Theorem 8.1. Theorem Let E be an open set in R d, and f a real-valued C 1 function on E. The following are equivalent: (i) The function f is locally operator monotone on E. (ii) The function f is in L d n(e) for all n. (iii) The function f is in L(E). 1.3 Local to Global In one variable, provided E is an interval, local monotonicity implies global monotonicity immediately. Indeed, suppose S T, and let S(t) = (1 t)s + tt. Then S (t) = T S 0, so f(t ) f(s) = 1 0 d f(s(t)) dt 0. (1.15) dt If E is not convex, this argument fails. Indeed, the function 1/x is locally n-matrix monotone on R\{0} for all n; but it is only globally monotone on sets that lie entirely on one side of 0. A result of Chandler [10] says that functions that are globally operator monotone on a set E always extend to be globally monotone on the convex hull of E. For intervals, (1.15) shows that the word locally can be dropped in both Theorem 1.5 and 1.7. One problem in going to several variables 5 is that this simple argument no longer works, because one may not be able to connect S and T by a path of commuting d-tuples. Indeed, the following example shows that there need not be any commuting tuples between two given ones. Example Let S and T be pairs in CSAM2 2 given by (( ) ( )) S =, (( ) ( )) T =, If R is in CSAM2 2 or R = T. and S R T, it can be shown that either R = S We have been unable to resolve the question of whether the n- matrix monotone functions on a connected open set E are a proper subset of the locally n-matrix monotone functions on E. However, as n tends to infinity and we pass to locally operator monotone functions, analyticity enters the picture, and makes the problem more tractable see Subsection The Nevanlinna Representation To prove (iii) (i) in Theorem 1.7, one must understand analytic functions that map the upper half-plane to itself. A key fact is a characterization due to R. Nevanlinna [26] which says that, provided they have some regularity at infinity, they are all Cauchy transforms of measures on the line. Theorem 1.17 (Nevanlinna). If F : Π Π is analytic and satisfies lim sup y y F (iy) C , for some C R, then there exists a unique finite positive Borel measure ν on R so that dν(t) F (z) = C + t z. (1.18) 6 Nevanlinna s theorem was used by M. Stone to prove the spectral theorem [36], but one can adopt the reverse viewpoint, and write (1.18) in terms of the resolvent of a self-adjoint. Indeed, let X be the self-adjoint operator of multiplication by the independent variable on L 2 (ν), and v the vector in L 2 (ν) that is 1 a.e. Then (1.18) can be rewritten as F (z) = C + (X z) 1 v, v. (1.19) This representation turns out to be useful in studying operator monotonicity, because then F (S) = CI + R v(i X S I) 1 R v, (1.20) where R v : H H M is given by R v : ξ ξ v. There is a several variable analogue of Theorem It may require first perturbing F. Definition For each real number t, define ρ t (z) = z + t 1 tz. For F L d, define F t by F t (z 1,..., z d ) := ρ t F (ρ t (z 1 ),..., ρ t (z d )). The following theorem follows from Theorem We shall say that a function F on Π d is analytic on a neighborhood of infinity if the function F (1/z 1,..., 1/z d ) extends to be analytic on a neighborhood of the origin. In Theorem 6.33, a weaker assumption is placed on F than being analytic in a neighborhood of infinity. Theorem Let F be in L d, and assume that F is analytic in a neighborhood of infinity. Then for all sufficiently small t, with at most countably many exceptions, the function F t has the following representation. There is a Hilbert space M, a densely defined selfadjoint operator X on M, a vector v in M, a real constant C, and d orthogonal projections P 1,..., P d with d P r = I M so that F t (z) = C + (X z r P r ) 1 v, v. (1.23) 7 1.5 Dimension d 2 : Global operator monotonicity If E is an open set in R d, we shall say that a real-valued function f defined on E is globally operator monotone, or just operator monotone for short, if, for every n, whenever S and T are in CSAM d n, with S T, and the joint eigenvalues of both S and T lie in E, then f(s) f(t ). Using the representation (1.23), we can prove results on (global) operator monotonicity. With notation as in Theorem 1.22, let us say that the µ-resolvent of X is the set of points {(z 1,..., z d ) C d : X z r P r has a bounded inverse}. We prove the following result as Theorem 9.2. Theorem Let X be a densely-defined self-adjoint operator on a Hilbert space M, let v be a vector in M, let C be a real constant, and let P 1,..., P d be projections with orthogonal ranges that sum to the identity. Let F be given by F (z) = C + (X z r P r ) 1 v, v. Let E be an open box in R d that is in the µ-resolvent of X. Then F is globally operator monotone on E. As an application, we can give a complete characterization of the rational functions of two variables that are operator monotone on rectangles. This is Theorem 9.6. Theorem Let F be a rational function of two variables. Let Γ be the zero-set of the denominator of F. Assume F is real-valued on R 2 \ Γ. Let E be an open rectangle in R 2 \ Γ. Then F is globally operator monotone on E if and only if F is in L(E), that is if and only if F is the restriction to E of an analytic function from Π 2 to Π that extends analytically across E. 8 2 Some Notation We shall let D denote the unit disk in the complex plane, Π the upper half-plane {z : Im (z) 0}, and H the right half-plane {z : Re (z) 0}. We shall let α(λ) = i 1 + λ (2.1) 1 λ be a linear fractional map that maps D to Π, and β(z) = z i z + i (2.2) be its inverse. We shall let d denote the number of variables. If z is a point in Π d, we shall use z 1,..., z d to denote its components; likewise λ = (λ 1,..., λ d ) will be a point in D d. We shall write S = (S 1,..., S d ) for a d-tuple of matrices or operators, and use S for max 1 r d S r. We shall also use α and β to denote the maps from D d to Π d and back again that are defined by applying α and β coordinate-wise. A kernel on a set E is a map K : E E C with the property that for every finite set {λ 1,..., λ N } of distinct points in E, the matrix [K(λ j, λ i )] is positive semi-definite. Definition 2.3. The Pick class, P d, is the set of analytic functions F : Π d Π. Definition 2.4. The Schur class, S d, is the set of analytic functions ϕ : D d D. Definition 2.5. The Carathéodory class, C d, is the set of analytic functions ψ : D d H. Definition 2.6. The Löwner class, L d, is the set of functions F : Π d Π with the property that there exist d kernel functions A r, 1 r d on Π d such that F (z) F (w) = (z 1 w 1 )A 1 (z, w) (z d w d )A d (z, w). (2.7) Definition 2.8. The Schur-Agler class, A d, is the set of functions ϕ : D d D with the property that there exist d kernel functions B r, 1 r d on D d such that 1 ϕ(λ)ϕ(µ) = (1 λ 1 µ 1 )B 1 (λ, µ) (1 λ d µ d )B d (λ, µ). (2.9) 9 When the dimension is clear, we shall drop the superscript d. Remark If we exclude the constant function 1 from S, we have the identification F P β F α S if α C. (2.11) Moreover, we also have (again excluding the constant function 1) F L β F α A, (2.12) (see Lemma 2.13). As all our results are trivial for constant functions, we shall use (2.11) and (2.12) without explicitly mentioning the exclusion of the constant function 1. The following change of variables formula is in [20]. A function is in A d if and only if it is analytic and maps d-tuples of commuting strict contractions to contractions; a function is in L d if and only if it is analytic and maps d-tuples of commuting operators with strictly positive imaginary parts 3 to operators with positive imaginary parts. Lemma The function F : Π d C is in the Löwner class if and only if ϕ := β F α is in the Schur-Agler class A d. Proof: Define ϕ = β F α. Then ϕ is in A d if and only if there are kernels B r on D such that 1 ϕ(λ)ϕ(µ) = (1 λ r µ r )B r (λ, µ). (2.14) When z = α(λ) and w = α(µ), (2.14) becomes 1 β F (z)β F (w) = ( 1 Rearranging (2.15), we get [ z r i z r + i ] [ w r ] ) i w r B r (β(z), β(w)). + i (2.15) F (z) F (w) = (z r w r ) F (z) + i z r + i F (w) i w r i Br (β(z), β(w)). (2.16) 3 We say an operator T has strictly positive imaginary part if there exists α 0 such that (T T )/2i αi. 10 If A r is defined for r = 1,..., d by (2.16) becomes A r (z, w) = F (z) + i F (w) i z r + i w r i Br (β(z), β(w)) F (z) F (w) = (z r w r )A r (z, w), (2.17) which means F is in L d. Reversing the argument gives the converse. Remark It is known that A d = S d for d = 1 or 2, and that for d 3 A d S d [12, 37, 2]. It follows similarly that the Löwner class equals the Pick class in dimensions 1 and 2, and is strictly contained in it for d 3. By Theorem of [31], rational inner functions are dense in the unit ball of S d in the topology of uniform convergence on compacta. Therefore there must be rational inner functions in S d \A d for d 3. By (2.11) and (2.12), it follows that for each d 3, there are rational functions that are real on R d and that are in P d \ L d. 3 Models, B-points and C-points For a function ϕ in A d, we can take the representation (2.9) and decompose the B r s as Gramians to get a Hilbert space model for ϕ. That means we find a separable Hilbert space M, an orthogonal decomposition of M, and an analytic map u : D d M such that 1 ϕ(µ)ϕ(λ) = M = M 1 M d, (3.1) (1 µ r λ r ) u r λ, ur µ M r (3.2) for all λ, µ D d, where we write u λ for u(λ), P r for the projection onto M r, and u r λ for P r [u λ ]. We shall view (3.1) interchangeably as a graded Hilbert space (i.e. one with a given orthogonal decomposition) or as a single Hilbert space 11 with d given projections P 1,..., P d that are orthogonal and add up to the identity. In general, if η M, we set η r = P r [η]. If λ C d, we may regard λ as an operator on M by letting Equation (3.2) can then be rewritten as λη = λ 1 η λ d η d. (3.3) 1 ϕ(µ)ϕ(λ) = (1 µ λ)u λ, u µ. (3.4) A lurking isometry argument yields the following result [2]. Theorem 3.5. If (M, u) is a model of ϕ A d, then there exist a C, vectors β, γ M and a linear operator D : M M such that the operator [ a 1 β ] γ 1 D is a contraction on C M and, for all λ D d, (1 Dλ)u λ = γ, (3.6) ϕ(λ) = a + λu λ, β. (3.7) With notation as in Theorem 3.5, we shall call (a, β, γ, D) a realization of (M, u). One can rewrite (3.6) and (3.7) as ϕ(λ) = a + β λ(i Dλ) 1 γ. (3.8) How one can go from (3.8) to (3.2) is discussed in [9] and [7]. If we start instead with the representation (2.7) of a function F in L d, we can decompose the kernels A r as the Grammians of some vectors v r, in auxiliary separable Hilbert spaces N r. Then we get, in the analogous notation to above, F (z) F (w) = = (z r w r )A r (z, w) (z r w r ) vz, r vw r N r = (z w )v z, v w N. (3.9) 12 This decomposition leads to a lurking self-adjoint argument, which we shall discuss in Section 6. In [4], we introduced the concept of a B-point for S. Let us give a unified definition for each of the classes S, P and C; notice that it depends on the codomain of the function. Definition Let U and V be fixed domains, and f : U V an analytic function. A point τ in U is called a B-point of f if there is a sequence λ n of points in U that converge to τ and such that dist(f(λ n ), V ) dist(λ n, U) (3.11) is bounded. So, for example, a point τ in Π d is a B-point for a function F in P d if there exists some sequence z n in Π d that tends to τ and such that the quantity Im F (z n ) min r {1,...,d} (Im zn) r is bounded. For a function in L d (respectively, A d ) we shall call a point τ a B-point if it is a B-point for the function thought of as an element of P d (resp. S d ). For each of the three classes S, P, and C, it follows from results of F. Jafari [19] and M. Abate [1] that if τ is a B-point, then the ratio (3.11) remains bounded for every sequence λ n that tends to τ nontangentially. Moreover, the function f will then have a non-tangential limit at τ.
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