Multipliers for Lipschitz p-Bessel sequences in metric spaces
aa r X i v : . [ m a t h . F A ] J u l MULTIPLIERS FOR LIPSCHITZ p-BESSEL SEQUENCES IN METRICSPACES
K. MAHESH KRISHNA AND P. SAM JOHNSONDepartment of Mathematical and Computational SciencesNational Institute of Technology Karnataka (NITK), SurathkalMangaluru 575 025, IndiaEmails: [email protected], [email protected]@gmail.com, [email protected]: July 8, 2020
Abstract : The notion of multipliers in Hilbert space was introduced by Schatten in1960 using orthonormal sequences and was generalized by Balazs in 2007 using Besselsequences. This was extended to Banach spaces by Rahimi and Balazs in 2010 using p-Bessel sequences. In this paper, we further extend this by considering Lipschitz functions.On the way we define frames for metric spaces which extends the notion of frames andBessel sequences for Banach spaces. We show that when the symbol sequence convergesto zero, the multiplier is a Lipschitz compact operator. We study how the variation ofparameters in the multiplier effects the properties of multiplier.
Keywords : Multiplier, Lipschitz operator, Lipschitz compact operator, frame, Besselsequence.
Mathematics Subject Classification (2020) : 42C15, 26A16.
Let { λ n } n ∈ ℓ ∞ ( N ) and { x n } n , { y n } n be sequences in a Hilbert space H . For x, y ∈ H ,the operator x ⊗ y is defined by x ⊗ y : H ∋ h
7→ h h, y i x ∈ H .The study of operators of the form ∞ X n =1 λ n ( x n ⊗ y n ) (1)began with Schatten [39], in connection with the study of compact operators. Schattenstudied the operator in (1) whenever { x n } n , { y n } n are orthonormal sequences in a Hilbertspace H . Later, operators in (1) are studied mainly in connection with Gabor analysis[7,14,18,21,22,41]. This was generalized by Balazs [5] who replaced orthonormal sequencesby Bessel sequences (we refer [11, 25] for Bessel sequences). Balazs and Stoeva studiedthese operators in [4, 42–44, 50–52]. 1et { f n } n be a sequence in the dual space X ∗ of a Banach space X and { τ n } n be a sequencein a Banach space Y . The operator τ ⊗ f is defined by τ ⊗ f : X ∋ x f ( x ) τ ∈ Y .It was Rahimi and Balazs [37] who extended the operator in (1) from Hilbert spaces toBanach spaces. For a Banach space X , and dual X ∗ , they considered the operator ∞ X n =1 λ n ( τ n ⊗ f n ) . (2)Rahimi and Balazs studied the operator in (2), whenever { τ n } n p-Bessel sequence (werefer [9,12] for p-Bessel sequences) for X ∗ and { f n } n q-Bessel sequence for X ( q is conjugateindex of p ). Besides theoretical importance, multipliers also play important role in frame(particularly Gabor) multipliers [21], signal processing [31], computational auditory sceneanalysis [53], sound synthesis [17], psychoacoustics [3], etc.In the present paper, we attempt to study the non-linear version of operator in (2). InSection 2 we recall necessary definitions and results which we use. In Section 3 we givedefinitions of frames for metric spaces, definition of multiplier for metric spaces and studythe properties of multiplier. In Hilbert spaces, a Riesz basis is defined as an image of an orthonormal basis under aninvertible operator [11]. In order to define Riesz basis for Banach spaces, one has to lookfor characterizations not involving inner product. Following is one such charaterization.
Theorem 2.1. [11] For a sequence { τ n } n in a Hilbert space H , the following are equiv-alent. (i) { τ n } n is a Riesz basis for H . (ii) span { τ n } n = H and there exist a, b > such that for every finite subset S of N , a X n ∈ S | c n | ! ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈ S c n τ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ b X n ∈ S | c n | ! , ∀ c n ∈ K . Definition 2.2. [2] Let < q < ∞ and X be a Banach space. A collection { τ n } n in X is said to be a q-Riesz sequence for X if there exist a, b > such that for every finite subset S of N , a X n ∈ S | c n | q ! q ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X n ∈ S c n τ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ b X n ∈ S | c n | q ! q , ∀ c n ∈ K . (3)(ii) q-Riesz basis for X if it is a q-Riesz sequence for X and span { τ n } n = X . We now recall the definition of a frame for a Hilbert space.
Definition 2.3. [11] A collection { τ n } n in a Hilbert space H is said to be a frame for H if there exist a, b > such that a k h k ≤ ∞ X n =1 |h h, τ n i| ≤ b k h k , ∀ h ∈ H . By realizing that the functional
H ∋ h
7→ h h, τ n i ∈ K is bounded linear, Definition 2.3leads to the following in Banach spaces. Definition 2.4. [2, 12] Let < p < ∞ and X be a Banach space. (i) A collection { f n } n of bounded linear functionals in X ∗ is said to be a p-frame for X if there exist a, b > such that a k x k ≤ ∞ X n =1 | f n ( x ) | p ! p ≤ b k x k , ∀ x ∈ X . (ii) A collection { τ n } n in X is said to be a p-frame for X ∗ if there exist a, b > suchthat a k f k ≤ ∞ X n =1 | f ( τ n ) | p ! p ≤ b k f k , ∀ f ∈ X ∗ . For more about p-frames for Banach spaces we refer [9, 45–49].We now recall the definition of Lipschitz function. Let M , N be metric spaces. A function f : M → N is said to be Lipschitz if there exists b > d ( f ( x ) , f ( y )) ≤ b d ( x, y ) , ∀ x, y ∈ M . Definition 2.5. [54] Let X be a Banach space. Let M be a metric space. The collection Lip( M , X ) is defined as Lip( M , X ) := { f : f : M → X is Lipschitz } . For f ∈ Lip( M , X ) , the Lipschitz number is defined as Lip( f ) := sup x,y ∈M ,x = y k f ( x ) − f ( y ) k d ( x, y ) . (ii) Let ( M , be a pointed metric space. The collection Lip ( M , X ) is defined as Lip ( M , X ) := { f : f : M → X is Lipschitz and f (0) = 0 } . For f ∈ Lip ( M , X ) ,the Lipschitz norm is defined as k f k Lip := sup x,y ∈M ,x = y k f ( x ) − f ( y ) k d ( x, y ) . Theorem 2.6. [54] Let X be a Banach space. (i) If M is a metric space, then Lip( M , X ) is a semi-normed vector space w.r.t. thesemi-norm Lip( · ) . (ii) If ( M , is a pointed metric space, then Lip ( M , X ) is a Banach space w.r.t. thenorm k · k Lip . The spaces Lip( M , X ) and Lip ( M , X ) are well-studied and we refer [13, 23, 30, 35, 54] forfurther information.In the theory of bounded linear operators between Banach spaces, an operator is said tobe compact if the image of the unit ball under the operator is precompact [20]. Linearityof the operator now gives various charaterizaions of compactness and plays importantrole in rich theories such as theory of integral equations [16], spectral theory [6], theory ofFredholm operators [19], operator algebra (C*-algebra) [15], K-theory [38], Calkin algebra[8], (operator) ideal theory [36], approximation properties of Banach spaces [28], Schauderbasis theory [28]. Lack of linearity is a hurdle when one tries to define compactness ofnon-linear maps. This hurdle was sucessefully crossed in the paper which began the studyof Lipschitz compact operators. We now record these things which are necessary in thepaper. Definition 2.7. [27] If M is a metric space and X is a Banach space, then the Lipschitzimage of a Lipschitz map (also called as Lipschitz operator) f : M → X is defined as theset (cid:26) f ( x ) − f ( y ) d ( x, y ) : x, y ∈ M , x = y (cid:27) . (4)4e observe that whenever an operator is linear, the set in (4) is simply the image of theunit sphere. Definition 2.8. [27] If ( M , is a pointed metric space and X is a Banach space, thena Lipschitz map f : M → X such that f (0) = 0 is said to be Lipschitz compact if itsLipschitz image is relatively compact in X , i.e., the closure of the set in (4) is compactin X . As showed in [27], there is a large collection of Lipschitz compact operators. To statethis, first we need a definition.
Definition 2.9. [10] Let ( M , be a pointed metric space and X be a Banach space.A Lipschitz operator f : M → X such that f (0) = 0 is said to be strongly Lipschitzp-nuclear ( ≤ p < ∞ ) if there exist operators A ∈ B ( ℓ p ( N ) , X ) , g ∈ Lip ( M , ℓ ∞ ( N )) and a diagonal operator M λ ∈ B ( ℓ ∞ ( N ) , ℓ p ( N )) induced by a sequence λ ∈ ℓ p ( N ) such that f = AM λ g , i.e., the following diagram commutes. M X ℓ ∞ ( N ) ℓ p ( N ) fg M λ A Proposition 2.10. [27] Every strongly Lipschitz p-nuclear operator from a pointed metricspace to a Banach space is Lipschitz compact.
Since the image of a linear operator is a subspace, the natural definition of finite rankoperator is that image is a finite dimensional subspace. The image of Lipschitz map maynot be a subspace. Thus care has to be taken while defining rank of such maps.
Definition 2.11. [27] If ( M , is a pointed metric space and X is a Banach space,then a Lipschitz function f : M → X such that f (0) = 0 is said to have Lipschitz finitedimensional rank if the linear hull of its Lipschitz image is a finite dimensional subspaceof X . Definition 2.12. [27] If M is a metric space and X is a Banach space, then a Lipschitzfunction f : M → X is said to have finite dimensional rank if the linear hull of its imageis a finite dimensional subspace of X . Next theorem shows that for pointed metric spaces, Definitions 2.11 and 2.12 are equiva-lent. 5 heorem 2.13. [1, 27] Let ( M , be a pointed metric space and X be a Banach space.For a Lipschitz function f : M → X such that f (0) = 0 the following are equivalent. (i) f has Lipschitz finite dimensional rank. (ii) f has finite dimensional rank. (iii) There exist f , . . . , f n in Lip ( M , K ) and τ , . . . , τ n in X such that f ( x ) = n X k =1 f k ( x ) τ k , ∀ x ∈ M . In Hilbert spaces (and not in Banach spaces), every compact operator is approximable byfinite rank operators in the operator norm [20]. Following is the definition of approximableoperator for Lipschitz maps.
Definition 2.14. [27] If ( M , is a pointed metric space and X is a Banach space, thena Lipschitz function f : M → X such that f (0) = 0 is said to be Lipschitz approximableif it is the limit in the Lipschitz norm of a sequence of Lipschitz finite rank operators from M to X . Theorem 2.15. [27] Every Lipschitz approximable operator from pointed metric space ( M , to a Banach space X is Lipschitz compact. We first define the notion of frames for metric spaces.
Definition 3.1. (p-frame for metric space) Let ( M , d ) , ( N n , d n ) , ≤ n < ∞ be metricspaces. A collection { f n } n of Lipschitz functions, f n : M → N n is said to be a Lipschitzp-frame ( ≤ p < ∞ ) for M relative to {N n } n if there exist a, b > such that a d ( x, y ) ≤ ∞ X n =1 d n ( f n ( x ) , f n ( y )) p ! p ≤ b d ( x, y ) , ∀ x, y ∈ M . If a is allowed to take the value 0, then we say that { f n } n a Lipschitz p-Bessel sequencefor M . efinition 3.2. (p-frame for metric space w.r.t. scalars) Let M be a metric space. Acollection { f n } n of Lipschitz functions from M to K is said to be a Lipschitz p-frame for M if there exist a, b > such that a d ( x, y ) ≤ ∞ X n =1 | f n ( x ) − f n ( y ) | p ! p ≤ b d ( x, y ) , ∀ x, y ∈ M . Definition 3.3. (p-frame for a pointed metric space w.r.t. scalars) Let ( M , be a pointedmetric space. A collection { f n } n in Lip ( M , K ) is said to be a pointed Lipschitz p-framefor M if there exist a, b > such that a d ( x, y ) ≤ ∞ X n =1 | f n ( x ) − f n ( y ) | p ! p ≤ b d ( x, y ) , ∀ x, y ∈ M . Definition 3.4.
Let ( M , be a pointed metric space. A collection { τ n } n in M is saidto be a pointed Lipschitz p-frame for Lip ( M , K ) if there exist a, b > such that a k f − g k Lip ≤ ∞ X n =1 | f ( τ n ) − g ( τ n ) | p ! p ≤ b k f − g k Lip , ∀ f, g ∈ Lip ( M , K ) . Remark 3.5. (i)
Definition 3.1 even generalizes the notion of bi-Lipschitz embedding(Ribe program) of metric spaces (we refer [26, 32–34, 40] for more on bi-Lipschitzembedding) (in fact, we see this by taking a fixed point z ∈ N and defining f n ( x ) = z, ∀ x ∈ N and ∀ n > ). It may happen that a metric space M may not embed inanother metric space N through bi-Lipschitz map. But it may have frames. We giveexamples to illustrate these things after this remark. (ii) By taking y = 0 and using f n (0) = 0 , for all n ∈ N , we see from Definition 3.2 that a d ( x, ≤ ∞ X n =1 | f n ( x ) | p ! p ≤ b d ( x, , ∀ x ∈ M . In particular, if M is a Banach space, then a k x k ≤ ∞ X n =1 | f n ( x ) | p ! p ≤ b k x k , ∀ x ∈ M . imilarly by taking g = 0 in Definition 3.4, we see that a k f k Lip ≤ ∞ X n =1 | f ( τ n ) | p ! p ≤ b k f k Lip , ∀ f ∈ Lip ( M , K ) . (iii) If M is a Banach space and f n ’s are all bounded linear functionals, then Definition3.1 becomes (i) in Definition 2.3. (iv) Since we only want the definition of Lipschitz p-Bessel sequence, we do not addressfurther properties of Lipschitz frames for metric spaces in this paper. However, wemake a detailed study of frames for metric spaces in [29].
Let x, y be distinct reals and consider M = { x, y } as a metric subspace of R . Then M does not embed in metric spaces N = { x } or N = { y } . Define f ( x ) = f ( y ) = x and f ( x ) = f ( y ) = y . Then f and f are Lipschitz and | f ( x ) − f ( y ) | p + | f ( x ) − f ( y ) | p =2 | x − y | p . Hence { f , f } is a Lipschitz p-frame for M .As another example, consider m, n ∈ N with m < n . Since a bi-Lipschitz map is injectiveand continuous, and there is no continuous injection from R n to R m (Corollary 2B.4 in [24])it follows that R n cannot be embedded in R m . Now define f j : R n ∋ ( x , . . . , x n ) ( x j , x j +1 , . . . , x n , x , . . . , x m − n + j − ) ∈ R m for 1 ≤ j ≤ n . Then f j is Lipschitz for all1 ≤ j ≤ n and n X j =1 k f j ( x , . . . , x n ) − f j ( y , . . . , y n ) k p = m k ( x , . . . , x n ) − ( y , . . . , y n ) k p , ∀ ( x , . . . , x n ) , ( y , . . . , y n ) ∈ R n . Thus { f , f , . . . , f n } is a Lipscitz p-frame for R n .We next give an example which uses infinite number of Lipschitz functions. Let 1 < a
Theorem 3.7.
Let { f n } n in Lip ( M , K ) be a pointed Lipschitz p-Bessel sequence for apointed metric space ( M , with bound b and { τ n } n in a Banach space X be a pointedLipschitz q-Bessel sequence for Lip ( X , K ) with bound d . If { λ n } n ∈ ℓ ∞ ( N ) , then the map T : M ∋ x ∞ X n =1 λ n ( τ n ⊗ f n ) x ∈ X is a well-defined Lipschitz operator such that T with Lipschitz norm at most bd k{ λ n } n k ∞ . roof. Let n, m ∈ N with n ≤ m . Then for each x ∈ M , using Holder’s inequality, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X k = n λ k ( τ k ⊗ f k )( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X k = n λ k f k ( x ) τ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup φ ∈X ∗ , k φ k≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ m X k = n λ k f k ( x ) τ k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup φ ∈X ∗ , k φ k≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k = n λ k f k ( x ) φ ( τ k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup φ ∈X ∗ , k φ k≤ m X k = n | λ k || f k ( x ) || φ ( τ k ) |≤ sup n ∈ N | λ n | sup φ ∈X ∗ , k φ k≤ m X k = n | f k ( x ) || φ ( τ k ) |≤ sup n ∈ N | λ n | sup φ ∈X ∗ , k φ k≤ m X k = n | f k ( x ) | p ! p m X k = n | φ ( τ k ) | q ! q ≤ sup n ∈ N | λ n | sup φ ∈X ∗ , k φ k≤ m X k = n | f k ( x ) | p ! p d k φ k = d sup n ∈ N | λ n | m X k = n | f k ( x ) | p ! p . Since ( P ∞ k =1 | f k ( x ) | p ) p converges, P ∞ k =1 λ k ( τ k ⊗ f k )( x ) also converges. Now for all x, y ∈M , 10 T x − T y k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 λ n f n ( x ) τ n − ∞ X n =1 λ n f n ( y ) τ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 λ n ( f n ( x ) − f n ( y )) τ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup φ ∈X ∗ , k φ k≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ ∞ X n =1 λ n ( f n ( x ) − f n ( y )) τ k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup φ ∈X ∗ , k φ k≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 λ n ( f n ( x ) − f n ( y )) φ ( τ k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup n ∈ N | λ n | sup φ ∈X ∗ , k φ k≤ ∞ X n =1 | f n ( x ) − f n ( y ) | p ! p ∞ X n =1 | φ ( τ n ) | q ! q ≤ sup n ∈ N | λ n | sup φ ∈X ∗ , k φ k≤ ∞ X n =1 | f n ( x ) − f n ( y ) | p ! p d k φ k = d sup n ∈ N | λ n | ∞ X n =1 | f n ( x ) − f n ( y ) | p ! p ≤ bd sup n ∈ N | λ n | d ( x, y ) . Hence k T k Lip = sup x,y ∈M ,x = y k T x − T y k d ( x, y ) ≤ bd sup n ∈ N | λ n | . Corollary 3.8.
Let { f n } n in Lip( M , K ) be a Lipschitz p-Bessel sequence for a metricspace M with bound b and { τ n } n in a Banach space X be a pointed Lipschitz q-Besselsequence for Lip ( X , K ) with bound d . If { λ n } n ∈ ℓ ∞ ( N ) , then for fixed z ∈ M , the map T : M ∋ x ∞ X n =1 λ n ( τ n ⊗ ( f n − f ( z ))) x ∈ X is a well-defined Lipschitz operator with Lipschitz number at most bd k{ λ n } n k ∞ . Proof.
Define g n := f n − f ( z ) , ∀ n ∈ N . Then for all x, y ∈ M , ( P ∞ n =1 | g n ( x ) − g n ( y ) | p ) p =( P ∞ n =1 | f n ( x ) − f n ( y ) | p ) p ≤ b d ( x, y ). Hence { g n } n is a Lipschitz p-Bessel sequence forpointed metric space ( M , z ) and we apply Theorem 3.7 to { g n } n . Definition 3.9.
Let { f n } n in Lip ( M , K ) be a pointed Lipschitz p-Bessel sequence fora pointed metric space ( M , and { τ n } n in a Banach space X be a pointed Lipschitz -Bessel sequence for Lip ( X , K ) . Let { λ n } n ∈ ℓ ∞ ( N ) . The Lipschitz operator M λ,f,τ := ∞ X n =1 λ n ( τ n ⊗ f n ) is called as the Lipschitz ( p, q ) -Bessel multiplier. The sequence { λ n } n is called as symbolfor M λ,f,τ . We easily see that Definition 3.9 generalizes Definition 3.2 in [37]. By varying the symboland fixing other parameters in the multiplier we get map from ℓ ∞ ( N ) to Lip ( M , X ).Property of this map for Hilbert space was derived by Balazs (Lemma in [5]) and forBanach spaces it is due to Rahimi and Balazs (Proposition 3.3 in [37]). In the nextproposition we study it in the context of metric spaces. Proposition 3.10.
Let { f n } n in Lip ( M , K ) be a pointed Lipschitz p-Bessel sequencefor ( M , with non-zero elements, { τ n } n in X be a q-Riesz sequence for Lip ( X , K ) and { λ n } n ∈ ℓ ∞ ( N ) . Then the mapping T : ℓ ∞ ( N ) ∋ { λ n } n M λ,f,τ ∈ Lip ( M , X ) is a well-defined injective bounded linear operator.Proof. From the norm estimate of M λ,f,τ , we see that T is a well-defined bounded linearoperator. Let { λ n } n , { µ n } n ∈ ℓ ∞ ( N ) be such that M λ,f,τ = T { λ n } n = T { µ n } n = M µ,f,τ .Then P ∞ n =1 λ n f n ( x ) τ n = M λ,f,τ x = M µ,f,τ x = P ∞ n =1 µ n f n ( x ) τ n , ∀ x ∈ M ⇒ P ∞ n =1 ( λ n − µ n ) f n ( x ) τ n = 0, ∀ x ∈ M . Now using Inequality (3), a ∞ X n =1 | ( λ n − µ n ) f n ( x ) | q ! q ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 ( λ n − µ n ) f n ( x ) τ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 , ∀ x ∈ M = ⇒ ( λ n − µ n ) f n ( x ) = 0 , ∀ n ∈ N , ∀ x ∈ M . Let n ∈ N be fixed. Since f n = 0, there exists x ∈ M such that f n ( x ) = 0. Thereforewe get λ n − µ n = 0. By varying n ∈ N we arrive at λ n = µ n , ∀ n ∈ N . Hence T isinjective.The result that a norm-limit of finite rank linear operators (between Banach spaces) is acompact operator [20] was generalized to Lipschitz operators in Theorem 2.15 by Jimenez-Vargas, Sepulcre, and Villegas-Vallecillos [27]. Using Theorem 2.15 we can generalizeLemma 3.6 in [37]. 12 roposition 3.11. Let { f n } n in Lip ( M , K ) be a pointed Lipschitz p-Bessel sequencefor ( M , with bound b and { τ n } n in X be a pointed Lipschitz q-Bessel sequence for Lip ( X , K ) with bound d . If { λ n } n ∈ c ( N ) , then M λ,f,τ is a Lipschitz compact operator.Proof. For each m ∈ N , define M λ m ,f,τ := P mn =1 λ n ( τ n ⊗ f n ). Then M λ m ,f,τ is a Lipschitzfinite rank operator (from Theorem 2.13). Now k M λ m ,f,τ − M λ,f,τ k Lip = sup x,y ∈M ,x = y k ( M λ m ,f,τ − M λ,f,τ ) x − ( M λ m ,f,τ − M λ,f,τ ) y k d ( x, y )= sup x,y ∈M ,x = y (cid:13)(cid:13)P ∞ n = m +1 λ n f n ( x ) τ n − P ∞ n = m +1 λ n f n ( y ) τ n (cid:13)(cid:13) d ( x, y )= sup x,y ∈M ,x = y (cid:13)(cid:13)P ∞ n = m +1 λ n ( f n ( x ) − f n ( y )) τ n (cid:13)(cid:13) d ( x, y ) ≤ bd sup m +1 ≤ n< ∞ | λ n | → m → ∞ . Hence M λ,f,τ is the limit of a sequence of Lipschitz finite rank operators { M λ m ,f,τ } ∞ m =1 withrespect to the Lipschitz norm. Thus M λ,f,τ is Lipschitz approximable and from Theorem2.15 it follows that M λ,f,τ is Lipschitz compact.We now study the properties of multiplier by changing its parameters. These are knownas continuity properties of multipliers in the literature. Following result extends Theorem5.1 in [37]. Theorem 3.12.
Let { f n } n in Lip ( M , K ) be a pointed Lipschitz p-Bessel sequence for M with bound b and { τ n } n in X be a pointed Lipschitz q-Bessel sequence for Lip ( X , K ) withbound d and { λ n } n ∈ ℓ ∞ ( N ) . Let k ∈ N and let λ ( k ) = { λ ( k )1 , λ ( k )2 , . . . } , λ = { λ , λ , . . . } , τ ( k ) = { τ ( k )1 , τ ( k )2 , . . . } , τ kn ∈ X , τ = { τ , τ , . . . } . Assume that for each k , λ ( k ) ∈ ℓ ∞ ( N ) and τ ( k ) is a pointed Lipschitz q-Bessel sequence for Lip ( X , K ) . (i) If λ ( k ) → λ as k → ∞ in p-norm, then k M λ ( k ) ,f,τ − M λ,f,τ k Lip → as k → ∞ . (ii) If { λ n } n ∈ ℓ p ( N ) and P ∞ n =1 k τ ( k ) n − τ n k q → as k → ∞ , then k M λ,f,τ ( k ) − M λ,f,τ k Lip → as k → ∞ . roof. (i) Using Theorem 3.7, k M λ ( k ) ,f,τ − M λ,f,τ k Lip = sup x,y ∈M ,x = y k ( M λ ( k ) ,f,τ − M λ,f,τ ) x − ( M λ ( k ) ,f,τ − M λ,f,τ ) y k d ( x, y )= sup x,y ∈M ,x = y (cid:13)(cid:13)(cid:13)P ∞ n =1 ( λ ( k ) n − λ n ) f n ( x ) τ n − P ∞ n =1 ( λ ( k ) n − λ n ) f n ( y ) τ n (cid:13)(cid:13)(cid:13) d ( x, y )= sup x,y ∈M ,x = y (cid:13)(cid:13)(cid:13)P ∞ n =1 ( λ ( k ) n − λ n )( f n ( x ) − f n ( y )) τ n (cid:13)(cid:13)(cid:13) d ( x, y ) ≤ bd sup n ∈ N | λ ( k ) n − λ n | = bd k{ λ ( k ) n − λ n } n k ∞ ≤ bd k{ λ ( k ) n − λ n } n k p → k → ∞ . (ii) Using Holder’s inequality, k M λ,f,τ ( k ) − M λ,f,τ k Lip = sup x,y ∈M ,x = y k ( M λ,f,τ ( k ) − M λ,f,τ ) x − ( M λ,f,τ ( k ) − M λ,f,τ ) y k d ( x, y )= sup x,y ∈M ,x = y (cid:13)(cid:13)(cid:13)P ∞ n =1 λ n f n ( x )( τ ( k ) n − τ n ) − P ∞ n =1 λ n f n ( y )( τ ( k ) n − τ n ) (cid:13)(cid:13)(cid:13) d ( x, y )= sup x,y ∈M ,x = y (cid:13)(cid:13)(cid:13)P ∞ n =1 λ n ( f n ( x ) − f n ( y ))( τ ( k ) n − τ n ) (cid:13)(cid:13)(cid:13) d ( x, y )= sup x,y ∈M ,x = y sup φ ∈X ∗ , k φ k≤ (cid:12)(cid:12)(cid:12)P ∞ n =1 λ n ( f n ( x ) − f n ( y )) φ ( τ ( k ) n − τ n ) (cid:12)(cid:12)(cid:12) d ( x, y ) ≤ sup x,y ∈M ,x = y sup φ ∈X ∗ , k φ k≤ ( P ∞ n =1 | λ n ( f n ( x ) − f n ( y )) | p ) p (cid:16)P ∞ n =1 | φ ( τ ( k ) n − τ n ) | q (cid:17) q d ( x, y ) ≤ sup x,y ∈M ,x = y sup φ ∈X ∗ , k φ k≤ ( P ∞ n =1 | λ n | p ) p ( P ∞ n =1 | f n ( x ) − f n ( y ) | p ) p (cid:16)P ∞ n =1 | φ ( τ ( k ) n − τ n ) | q (cid:17) q d ( x, y ) ≤ b k{ λ n } n k p ∞ X n =1 k τ ( k ) n − τ n k q ! q → k → ∞ . Acknowledgments
The first author thanks the National Institute of Technology Karnataka (NITK), Surathkalfor giving him financial support and the present work of the second author was partiallysupported by National Board for Higher Mathematics (NBHM), Ministry of Atomic En-ergy, Government of India (Reference No.2/48(16)/2012/NBHM(R.P.)/R&D 11/9133).
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