Lebesgue Spaces of Matrix Functions

Transcription

1 Chapter 2 Lebesgue Spaces of Matrix Functions In this Chapter, we introduce the notations and define the spaces L p (,M n ) of matrix L p functions on locally compact groups as a setting for later developments. We recall some basic definitions and derive some results for convolution operators in the scalar case. We discuss differentiability of the norm in L p (,M n ) which is needed later, and compute the ateaux derivative of the norm when the matrix space M n is equipped with the Hilbert-Schmidt norm. 2.1 Preliminaries We denote by throughout a locally compact group with identity e and a right invariant Haar measure λ. To avoid the inconvenience of additional measure-theoretic technicalities, we assume throughout that λ is σ-finite. If is compact, λ is normalized to λ()=1. Let 1 p <. iven a complex Banach space E, we denote by L p (,E) the Banach space of (equivalence classes of) E-valued Bochner integrable functions f on satisfying ( 1 f p = f (x) dλ(x)) p p < (cf. [22, p.97]). We write L p () for L p (,E) if dime = 1. In the sequel, E is usually the C -algebra M n of n n complex matrices in which case, a function f : M n is an n n matrix ( f ij ) of complex functions f ij on. We denote by B(E) the Banach algebra of bonded linear self-maps on a Banach space E. Let Tr : M n C be the canonical trace of M n. Every continuous linear functional ϕ : M n C is of the form ϕ( )=Tr( A ϕ ) where the matrix A ϕ M n is unique and ϕ = Tr( A ϕ )=Tr((A ϕa ϕ ) 1/2 ) which is the trace-norm A ϕ tr of A ϕ. We will identify the dual Mn, via the map ϕ Mn A ϕ M n, with the vector space M n equipped with the trace-norm tr. If we equip M n with the Hilbert-Schmidt norm C.-H. Chu, Matrix Convolution Operators on roups. Lecture Notes in Mathematics 1956, 5 doi: / , c Springer-Verlag Berlin Heidelberg 2008

2 6 2 Lebesgue Spaces of Matrix Functions A hs = Tr(A A) 1/2, then M n is a Hilbert space with inner product A,B = Tr(B A). We note that the C*-norm, the trace-norm and the Hilbert-Schmidt norm on M n are related by tr n hs n and norm convergence is equivalent to entry-wise convergence in M n. If M n is equipped with the Hilbert-Schmidt norm, then L 2 (,(M n, hs )) is a Hilbert space, with inner product f,g 2 = Tr( f (x)g(x) )dλ(x). Since f (x)g(x) hs f (x) hs g(x) hs for f,g L 2 (,(M n, hs )), the Bochner integral f,g = f (x)g(x) dλ(x) exists in M n and defines an M n -valued inner product, turning L 2 (,(M n, hs )) into an inner product (left) M n -module. We denote by L (,M n ) the complex Banach space of M n -valued essentially bounded (locally) λ-measurable functions on, where M n is equipped with the C*- norm. It is a von Neumann algebra, with predual L 1 (,Mn), under the pointwise product and involution: ( fg)(x)= f (x)g(x), f (x)= f (x) ( f,g L (,M n ), x ). We will study convolution operators on L p (,M n ) defined by matrix-valued measures. In this section, we first recall some basic definitions and derive some results for convolution operators on L p (), for later reference. One important difference in the matrix setting is the presence of non-commutative and non-associative algebraic structures. We equip the vector space C() of complex continuous functions on with the topology of uniform convergence on compact sets in, and denote by C c () the subspace of functions with compact support. The Banach space of bounded complex continuous functions on is denoted by C b (). LetC 0 () be the Banach space of complex continuous functions on vanishing at infinity. The dual C 0 () identifies with the space M() of complex regular Borel measures on. Each µ M() has finite total variation µ and M() is a unital Banach algebra in the total variation norm and the convolution product: µ = µ (), f, µ ν = f (xy)dµ(x)dν(y) (f C 0 (), µ,ν M()) where we always denote the duality of a dual pair of Banach spaces E and F by, : E F C.

3 2.1 Preliminaries 7 We also write µ( f ) for f, µ = fdµ. The unit mass at a point a is denoted by δ a where δ e is the identity in M(). A measure µ M() is called absolutely continuous if its total variation µ is absolutely continuous with respect to the Haar measure λ. iven σ M(),thesupport of σ is defined to be the support of its total variation σ and is denoted by suppσ. We denote by σ the closed subgroup of generated by the support of σ. A measure σ M() is called adapted if σ =. A measure σ M() is said to be non-degenerate if supp σ generates a dense semigroup in. Evidently, every non-degenerate measure is adapted. An absolutely continuous (non-zero) measure on a connected group must be adapted. By a (complex) measure µ on, we will mean a measure µ M()\{0}. The convolutions for Borel functions f and g on, when exit, are defined by ( f g)(x) = f (xy 1 )g(y)dλ(y); ( f µ)(x) = f (xy 1 )dµ(y); (µ f )(x) = f (y 1 x) (y 1 )dµ(y) where is the modular function satisfying dλ(xy)= (x)dλ(y) and dλ(x 1 )= (x 1 )dλ(x). We denote by l x and r x, respectively, the left and right translations by an element x : l x f (y)= f (x 1 y), r x f (y)= f (yx) (y ) for any function f on. A complex function f on is left uniformly continuous if r x f f 0asx e. It is right uniformly continuous if l x f f 0as x e. We also write x f = l x 1 f and f x for r x f. We note that each f C c () is both left and right uniformly continuous, and for any µ M(),wehave f µ C b () since f µ(x) f µ(y) l xy 1 f f µ. We also have f, µ ν = f, ν µ (2.1) where ν M() and we define f (x)= f (x 1 ) and d µ(x)=dµ(x 1 ). Note that µ( f )=µ( f )=(f µ)(e) and µ ν = ν µ for f C c (). Let σ M().For1 p, we define the convolution operator T σ : L p () L p () by T σ ( f )= f σ ( f L p ()). To avoid triviality, σ is always non-zero for T σ. The definition of T σ depends on its domain L p () although we often omit referring to it if there is no ambiguity. When regarded as an operator on L p (), the operator T σ is easily seen to be bounded and we denote its norm by T σ p,orsimply T σ in obvious context. We have T σ p σ.

4 8 2 Lebesgue Spaces of Matrix Functions A convolution operator T σ : L p () L p () commutes with left translations: l x T σ = T σ l x (x ). Conversely, for abelian groups, every translation invariant operator T : L 1 () L 1 () is a convolution operator T σ for some σ M() [55, 3.8.4]. However, this result does not hold for 1 < p, evenif is compact and abelian [44, p.85]. We will characterise the more general matrix convolution operators in Chapter 3. In particular, the above L 1 result is generalized to the matrix-valued case, for all locally compact groups. For 1 p, we denote by q its conjugate exponent throughout, that is, 1 p + 1 q = 1, and for the dual pairing, between Lp () and L q (),wehave f σ,h = f,h σ (2.2) for f L p () and h L q (). This implies that T σ is weakly continuous on L p () for 1 p <, and is weak* continuous on L (). In particular, T σ is a weakly compact operator on L p () for 1 < p <. Forp = 1,, we will discuss presently weak compactness of T σ : L p () L p (), but we note the following two lemmas first. Lemma Let σ M() and p <. Let T σ : L q () L q () be the dual map of the convolution operator T σ : L p () L p (). Then T σ = T σ. The operator T σ : L 2 () L 2 () is self-adjoint if σ = σ is a real measure. The weak* continuous operator T σ : L () L () has predual T σ : L 1 () L 1 (). Proof. By (2.2), we have f,tσ h = f,t σ h for f L p () and h L q (). The adjoint of T σ in B(L 2 ()) is where σ is the complex conjugate of σ. T σ Lemma Let σ M() and let T σ be the convolution operator on L p () for p = 1,. We have T σ 1 = T σ = σ. Proof. We have σ = sup{ fdσ : f C c() and f 1} in which fdσ = f σ(e) f σ T σ where f σ C b (). Next, we have T σ 1 = T σ = T σ = σ = σ. Remark We note that T σ p need not equal σ if 1 < p <. Indeed, if σ is an adapted probability measure whose support contains the identity e and if T σ p = 1forsome1< p <, then is amenable (see, for example, [4, Theorem 1]). On the other hand, if is amenable and σ is a probability measure, then T σ p = 1for all p (cf. [33, p.48]). By Lemma 2.1.2, the spectral radius of T σ B(L p ()), for p = 1,, is lim n T n σ 1 n = lim n T σ n 1 n = lim n σ n 1 n where σ n is the n-fold convolution of σ with itself.

5 2.1 Preliminaries 9 Lemma Let be a compact group and let σ M() be absolutely continuous. Then the convolution operator T σ : L p () L p () is compact for every p [1, ]. Proof. Let σ = h λ for some h L 1 (). Consider first T σ : L () L (). By absolute continuity of σ,wehavet σ (L ()) C(). Hence, by Arzela-Ascoli theorem, we need only show that the set {T σ ( f ) : f 1} is equicontinuous in C(). Let ε > 0. Pick ϕ C c () with support K and ϕ h 1 < 4 ε.letw be a compact neighbourhood of the identity e. By uniform continuity, we can choose a compact neighbourhood V W of e such that whenever x 1 y V. Then ϕ x ϕ y 1 = = ϕ(x) ϕ(y) < KW ε 2λ(KW) ϕ(zx) ϕ(zy) dλ(z) ϕ(z) ϕ(zx 1 y) dλ(z) < ε 2. It follows that, for x 1 y V and f 1, we have T σ ( f )(x) T σ ( f )(y) = f (xz 1 )h(z)dλ(z) f (yz 1 )h(z)dλ(z) f (z 1 )h(zx) f (z 1 )h(zy) dλ(z) f h x h y 1 f ( h x ϕ x 1 + ϕ x ϕ y 1 + h y ϕ y 1 ) < ε which proves equicontinuity and hence, compactness of T σ : L () L (). Likewise T σ : L () L () is compact and therefore T σ : L 1 () L 1 () is compact. Let 1 < p <. Let(h n ) be a sequence in C() such that h n h 1 0. Then T σ = lim n T σn in B(L p ()), where σ n = h n λ. Hence it suffices to show compactness of T σ on L p () for the case h C(). Let ( f n ) be a sequence in the unit ball of L p (). Then f n 1 1 for all n and compactness of T σ : L 1 () L 1 () implies that the sequence ( f n σ) contains a subsequence L 1 -converging to some f L 1 (), and hence a subsequence ( f k σ) converging pointwise to f λ-almost everywhere. Since h C(), wehave f k σ f k p h q h q for all k, and f L ().Itfollowsthat

6 10 2 Lebesgue Spaces of Matrix Functions f k σ f p p f k σ f 1 f k σ f p 1 0 ask. This proves compactness of T σ : L p () L p (). Remark The above result is clearly false if σ is not absolute continuous, for instance, T σ is the identity operator if σ = δ e. A compactness criterion has been given in [48] for a class of convolution operators of the form f L 1 () f F C() where F L () and is compact abelian. Compactness of the composition of a convolution operator with a multiplier has also been considered in [59, 60]. Fredholmness of convolution operators on locally compact groups has been studied in [54, 59, 61]. Proposition Let σ be a positive measure on a group such that σ 2 σ 2 is adapted. Let T σ be the associated convolution operator. The following conditions are equivalent. (i) T σ : L 1 () L 1 () is weakly compact. (ii) T σ : L 1 () L 1 () is compact. (iii) T σ : L () L () is weakly compact. (iv) T σ : L () L () is compact. (v) T σ : L p () L p () is compact for all p [1, ]. (vi) is compact and σ is absolutely continuous. Proof. (i) = (vi). We first prove compactness of. Note that L 1 () has the Dunford-Pettis property and in particular, every weakly compact operator on L 1 () sends weakly compact subsets to norm compact sets [22, p.154]. Hence weak compactness of T σ implies that the operator Tσ 2 : L 1 () L 1 () is compact, and so is the operator T σ σ σ σ = T 2 σ T σ 2. Since σ 2 σ 2 is a positive measure, the spectral radius of T σ 2 σ 2 is σ()4, by a remark before Lemma On the Hilbert space L 2 (), the operator T σ 2 σ 2 = T σ 2 T σ 2 is a positive operator and therefore has only non-negative eigenvalues. The eigenvalues of T σ 2 σ 2 B(L1 ()) are also eigenvalues of T σ 2 σ 2 B(L2 ()) and therefore non-negative. It follows that σ() 4 is an eigenvalue of the compact operator T σ 2 σ 2 B(L1 ()), that is, there is a nonzero function f L 1 () satisfying f σ 2 σ 2 = σ() 4 f. Note that the measure σ() 4 σ 2 σ 2 is an adapted probability measure on. Now, by [10, Theorem 3.12], f is constant which implies that must be compact. Next, we show that σ is absolutely continuous. By the Dunford-Pettis-Phillips Theorem [22, p.75], there is an essentially bounded function g : L 1 () such that T σ ( f )= fgdλ ( f L 1 ()). Now the arguments in [22, p.91] can still be applied without commutativity of. Let a. For each f L 1 (),wehave,forλ-a.e. y,

7 2.1 Preliminaries 11 f (x)g(x)(y)dλ(x) =T σ f (y)=l a 1T σ (l a f )(y) = (l a f )(x)g(x)(ay)dλ(x) = f (a 1 x)g(x)(ay)dλ(x) = f (x)g(ax)(ay)dλ(x). It follows that g(ax)(ay)=g(x)(y) for λ-a.e. x and y. This implies that, for each f C(), the function F(y)= f (x)g(yx 1 )(y)dλ(x) (y ) is invariant under the left translations l a for all a. Using compactness of, one can show that F is constant λ-almost every on, as in [22, p.91], and hence we have, F(y)= F(z)dλ(z)= f (x)g(zx 1 )(x)dλ(x)dλ(z) (2.3) for λ-a.e. y. Leth L 1 () be defined by h(x)= g(yx 1 )(y)dλ(y). We show that f σ = f h for each f C() L 1 () which then yields absolutely continuity of σ. Indeed, for each k L (),wehave k, f h = = = = k(y) k(y) k(y) f (yx 1 )h(x)dλ(x)dλ(y) f (yx 1 )g(zx 1 )(z)dλ(x)dλ(z)dλ(y) f (yx 1 )g(yx 1 )(y)dλ(x)dλ(y) (by (2.3)) k(y) f σ(y)dλ(y)= k, f σ which concludes the proof. (vi) = (v). By Lemma (v) = (iv) = (iii). Trivial. (iii) = (ii). The given condition implies that T σ : L 1 ) L 1 () is weakly compact. Repeating (i) = (v) = (iv) for σ, we see that T σ : L () L () is compact, and hence T σ : L 1 () L 1 () is compact. (ii) = (i). Trivial.

8 12 2 Lebesgue Spaces of Matrix Functions Remark In (i) = (vi) above, the proof of absolute continuity of σ from weak compactness of T σ B(L 1 ()) is valid for any measure σ on a compact group, without adaptedness of σ 2 σ 2. Corollary iven a positive absolutely continuous measure σ on a connected group, the following conditions are equivalent. (i) T σ : L 1 () L 1 () is weakly compact. (ii) T σ : L () L () is weakly compact. (iii) T σ : L p () L p () is compact for all p [1, ]. (iv) is compact. Proof. This is because absolutely continuous measures on a connected group are adapted. Definition The spectrum of an element a in a unital Banach algebra A is denoted by Spec A a which is often shortened to Speca if the Banach algebra A is understood. For 1 p, we write Spec(T σ,l p ()), or simply, Spec(T σ,l p ),for the spectrum SpecT σ, when regarding T σ B(L p ()). We denote by Λ(T σ,l p ()), or simply, Λ(T σ,l p ), the set of eigenvalues of T σ : L p () L p (). iven any Banach algebra A and an element a A, we define, as usual, the quasi-spectrum of a, denoted by Spec A a, to be the spectrum Spec A 1 a of a in the unit extension A 1 of A. We always have 0 Spec A a.ifahas an identity, then we have Spec A a = Spec A a {0}. We recall that Spec(T σ,l p )=Λ(T σ,l p ) Spec r (T σ,l p ) Spec c (T σ,l p ) where Spec r (T σ,l p ) denotes the residue spectrum of T σ, consisting of α Spec(T σ,l p )\Λ(T σ,l p ) satisfying (T σ αi)(l p ()) L p () and Spec c (T σ,l p ) denotes the continuous spectrum of T σ, consisting of α Spec(T σ,l p )\Λ(T σ,l p ) such that Since T σ = T σ for p <, wehave (T σ αi)(l p ()) = L p (). Spec(T σ,l p )=Spec(T σ,l q ) for 1 p <, and also Spec(T σ,l )=Spec(T σ,l 1 ). We denote by Specσ the spectrum of σ in the measure algebra M(). Note that Specσ = Spec σ since σ µ = µ σ for each µ M().

9 2.1 Preliminaries 13 iven a locally compact group, weletĝ be the dual space consisting of (the equivalence classes of) continuous unitary irreducible representations π : B(H π ), where H π is a Hilbert space. Let ι Ĝ be the one-dimensional identity representation. For π Ĝ, σ M() and f L 1 (), we define the Fourier transforms: σ(π) = π(x 1 )dσ(x) B(H π ), f (π) = f (x)π(x 1 )dλ(x) B(H π ). We have f σ(π)= σ(π) f (π) and µ σ(π)= σ(π) µ(π) for µ M(). The spectrum Spec B(Hπ ) σ(π) of σ(π) B(H π) will be written as Spec σ(π) if no confusion is likely. If is abelian, Ĝ is the group of characters and we often use the letter χ to denote an element in Ĝ. For1< p < 2 and f L p (), we define the Fourier transform f L q (Ĝ) via Riesz-Thorin interpolation. A continuous homomorphism χ from an abelian group to the multiplicative group C\{0} is called a generalized character. For such a character χ with χ( ) 1, one can still define σ(χ) as above. The spectrum Ω() of the Banach algebra M(), i.e., the non-zero multiplicative functionals on M(), identifies with the generalized characters χ of with χ( ) 1, and by elfand theory, we have Specσ = σ(ω()) which contains σ(ĝ). The spectrum of L 1 () identifies with the dual group Ĝ and if is discrete, then M()=l 1 () and Specσ = Spec l 1 () σ = σ(ĝ). For arbitrary groups, we have the following result. Lemma Let σ be a complex measure on a group. Then Λ(T σ,l 1 ) Spec σ(π) Specσ. The inclusions are strict in general. π Ĝ Proof. Similar inclusions hold in the more general matrix setting for which a simple proof will be given in Proposition If σ is an adapted probability measure and is non-compact, then by [10, Theorem 3.12], 1 / Λ(T σ,l 1 ) while 1 Specι(σ) where ι Ĝ is the identity representation. If is abelian, then σ(ĝ)= π Ĝ Spec σ(π) and Example shows that the last inclusion can be strict. In fact, even the closure σ(ĝ) may not equal Specσ by Remark It has been shown in [10, Lemma 3.11] that 1 / π Ĝ\{ι} Spec σ(π) if σ is an adapted probability measure on a locally compact group. In general, there seem to be few definitive results concerning the spectrum of T σ for non-abelian groups. We will consider this case in Chapter 3 and prove various results there.

10 14 2 Lebesgue Spaces of Matrix Functions We will make use of a version of the Wiener-Levy theorem, stated below, which has been proved in [55, Theorem 6.2.4] and will be generalized to the matrix setting in Chapter 3. Lemma Let Ω be an open set in C and let F : Ω C be a real analytic function satisfying F(0) =0 if 0 Ω. iven an abelian group and a function f L 1 () such that f (Ĝ) Ω, then F( f ) is the Fourier transform of an L 1 ()- function. Example For the Cauchy distribution dσ t (x)= t π(t 2 + x 2 dx (t > 0) ) on R, we have σ t ( R) ={exp( t x ) : x R} =(0,1] =Spec(T σ,l p )\{0} = Λ(T σt,l ). Example Let be any locally compact group and let σ = δ a be the unit mass at a. Then T σ is a translation on L p () and we have Spec(T σ,l ) {α : α = 1}. If = T and a = i, then L (T) L 2 (T) and Spec(T σ,l )=Spec(T σ,l 2 )= σ(z)={exp( inπ/2) : n Z} = {±1,±i} {α : α = 1}. If = Z and a = 1, then Spec(T σ,l 2 )={α : α = 1} = Spec(T σ,l ). If = R and a 0, then Spec(T σ,l p )={α : α = 1} = Λ(T σ,l ) as δ a ( R)= {exp( iaθ) : θ R}. Next consider the measure µ = 1 2 (δ 0 + δ 1 ) on R. Its n-fold convolution µ n = 1 2 n n k=0 ( ) n δ k k is a convex sum of discrete measures and we have { Spec(T µ n,l )=Λ(T µ n,l n 1 )= 2 n k=0 ( ) n exp( ikθ) : θ R k where, for example, Spec(T µ,l ) is the circle containing 0 and internally tangent to the unit circle at 1, with sinπx as a 0-eigenfunction. } 2.2 Differentiability of Norm in L p (,M n ) We will be working with the complex Lebesgue spaces L p (,M n ) where, for convenience and consistency with previous and related works elsewhere, we equip M n with the C*-norm unless otherwise stated. Some remarks are in order here. First,

11 2.2 Differentiability of Norm in L p (,M n ) 15 there is no essential difference if one chooses to equip M n with the trace norm since it amounts to considering the space L p (,M n) which is, for p > 1, the dual of L q (,M n ). Also, the Lebesgue spaces L p (,M n ) defined in terms of the C*, trace and Hilbert-Schmidt norms on M n are all isomorphic and most results for these three cases are identical. There is, however, a difference among the three cases if one considers the differentiability of the norm of L p (,M n ) which will be needed later. Let us first consider the differentiability of the C*-norm, the trace norm tr and the Hilbert-Schmidt norm hs on M n, regarded as a real Banach space. We recall that the norm of a real Banach space E is said to be ateaux differentiable at a point u E if the following limit exists u (x)=lim t 0 u +tx u t for each x E, in which case, the limit is called the ateaux derivative of the norm at u, in the direction of x. We note that the right directional derivative always exists. In fact, it is equal to + u (x)=lim t 0 u +tx u t sup{ψ(x) : ψ is a subdifferential at u} where a linear functional ψ in the dual E is called a subdifferential at u if ψ(x u) x u for each x E. The norm is ateaux differentiable at u if, and only if, there is a unique subdifferential at u, in which case, the subdifferential is the ateaux derivative (cf. [53, Proposition 1.8]). The Hilbert-Schmidt norm hs on M n is ateaux differentiable at every A M n \{0}. Indeed, we have A +tx hs A hs lim t 0 t = lim t 0 Tr((A +tx) (A +tx)) Tr(A A) t( A +tx hs + A hs ) = Tr(A X + X A) 2 A hs = 1 A hs ReTr(A X). Although the norm of a separable Banach space is ateaux differentiable on a dense δ set, it is easy to see that the C*-norm and the trace norm need not be ateaux differentiable at every non-zero A M n. Lemma Let A M n \{0}. The C*-norm on M n is ateaux differentiable at A if, and only if, given any unit vectors ξ,η C n with Aξ = Aη = A, we have

12 16 2 Lebesgue Spaces of Matrix Functions Aξ,Xξ = Aη,Xη (X M n ). In the above case, the ateaux derivative at A is given by A (X)= 1 A Re Aξ,Xξ (X M n) where ξ C n is a unit vector satisfying Aξ = A. Proof. Suppose the norm is ateaux differentiable at A.Letξ C n be a unit vector such that A = Aξ. Define a real continuous linear functional ψ ξ : M n R by ψ ξ (X)= 1 Re Aξ,Xξ A (X M n). Then for each X M n,wehave ψ ξ (X A) = 1 Re Aξ,X Aξ A = 1 Re( Aξ,Xξ Aξ,Aξ ) X A. A Hence ψ ξ is a subdifferential at A. Ifη is a unit vector in C n such that Aη = A, then we must have ψ η = ψ ξ, by uniqueness of the subdifferential, which gives Aξ,Xξ = Aη,Xη for every X M n. To show the converse, we note that (cf. [5, Proposition 4.12]), { } A +tx A (A +tx)ξ Aξ lim = sup lim : ξ = 1, Aξ = A t 0 t t 0 t where (A +tx)ξ Aξ (A +tx)ξ,(a +tx)ξ Aξ,Aξ lim = lim t 0 t t 0 t( (A +tx)ξ + Aξ ) Aξ,Xξ + Xξ,Aξ =. 2 A Hence the necessary condition implies that the above set on the right reduces to a singleton which gives the right directional derivative. We also have (A +tx)ξ Aξ (A tx)ξ Aξ lim = lim t 0 t t 0 t Aξ, Xξ + Xξ,Aξ = 2 A (A +tx)ξ Aξ = lim. t 0 t

13 2.2 Differentiability of Norm in L p (,M n ) 17 This proves ateaux differentiability at A. The last assertion is clear from the above computation. ( ) 10 Example Let A = M Then the unit vectors in C 2 where A achieves its norm are of the form (α,0) with α = 1. For any matrix X =(x ij ) in M 2,we have A(α,0) T,X(α,0) T = x 12 + x 21 which is independent of α, and the C*-norm is ateaux differentiable at A with derivative The matrix B = A (X)=Re A(1,0) T,X(1,0) T = Rex 11. ( ) 11 achieves its norm at ( 2,0) and ( , 2 2 );but B( 2,0) T,X( ( 2 2,0) T B 2, ) T 2,X 2 ( 2 2, ) T 2 if X is the identity matrix, say. Hence the C*-norm is not ateaux differentiable at B, however, we have the right I-directional derivative + B +ti B B (I) =lim t 0 t = lim t 0 1 +t +t t + 2t 2 2 = t On the other hand, the trace norm tr is not ateaux differentiable at A since for X = ( ) 0 0,say. 0 1 A +tx tr A tr t = t t Lemma Let A M n \{0} with polar decomposition A = u A. If the trace norm tr on M n is ateaux differentiable at A, then the ateaux derivative is given by A tr (X)=ReTr(u X) (X M n ). Proof. We only need to show that ψ(x)=retr(u X) is a subdifferential. Indeed, we have A = u A and ψ(x A) =ReTr(u X) ReTr(u A) u X tr A tr = X tr A tr.

14 18 2 Lebesgue Spaces of Matrix Functions Example In Example 2.2.2( above, ) we have u = A in the polar decomposition 0 0 of A and ReTr(u X)=0forX =, while the right X-directional derivative 0 1 is given by A +tx tr A tr t lim = lim = 1. t 0 t t 0 t Due to the non-smoothness of the C*-norm and trace norm on M n, we will consider the Lebesgue spaces L p (,(M n, hs )) with M n equipped with the Hilbert- Schmidt norm when we need to make use of norm differentiability later. We compute below the ateaux derivatives for L p (,(M n, hs )). Since the function u E u p is convex on any Banach space E,wehave,for 0 < t < 1 and u,v E, u +tv p (1 t) u p +t u + v p and which gives u p t 1 +t u v p + 1 u +tv p 1 +t u p u v p 1 t ( u +tv p u p ) u + v p u p. (2.4) Proposition Let 1 < p <. The norm of L p (,(M n, hs )) is ateaux differentiable at each non-zero f with ateaux derivative f p (g)=re f 1 p p f (x) p 2 hs Tr( f (x) g(x))dλ(x) {x: f (x) 0} for g L p (,(M n, hs )). Proof. iven A M n \{0}, we have, by the chain rule, d dt A +tx p hs = d p A p 1 hs t=0 dt A +tx hs = p A p 1 hs ReTr(A X) t=0 for X M n. Fix a non-zero f in L p (,(M n, hs )).ivenp > 1 and g L p (,(M n, hs )), we have d dt tg(x) p hs = 0. t=0 By (2.4) and the dominated convergence theorem, we have

15 2.2 Differentiability of Norm in L p (,M n ) 19 p f p 1 d p dt f +tg p = d t=0 dt f +tg p p t=0 d = dt f (x)+tg(x) p hs dλ(x) t=0 d = {x: f (x) 0} dt f (x)+tg(x) p hs dλ(x) t=0 = p f (x) p 2 hs ReTr( f (x) g(x))dλ(x) {x: f (x) 0} which gives the formula for the ateaux derivative at f. Corollary For 1 < p <, the Lebesgue space L p (,(M n, hs )) is strictly convex, that is, the extreme points of its closed unit ball are exactly the functions of unit norm. Proof. This follows from the fact that a Banach space E is strictly convex if, and only if, the norm of its dual E is ateaux differentiable on the unit sphere.

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