A Hausdorff Operator on Lebesgue Space With Commuting Perturbation Matrices Is a Non-Riesz Operator
aa r X i v : . [ m a t h . F A ] N ov A HAUSDORFF OPERATOR ON LEBESGUE SPACE WITHCOMMUTING FAMILY OF PERTURBATION MATRICES ISA NON-RIESZ OPERATORA. R. Mirotin [email protected]
Abstract
We consider a generalization of Hausdorff operators on Lebesguespaces and under natural conditions prove that such an operator is not a Rieszoperator provided it is non-zero. In particular, it cannot be represented as a sumof a quasinilpotent and compact operators.
Key words and phrases. Hausdorff operator, Riesz operator, quasinilpo-tent operator, compact operator, Lebesgue space.MSC Class: 45P05; 47G10; 47B06
The one-dimensional Hausdorff operator( H f )( x ) = Z R f ( xt ) dχ ( t )( χ is a measure supported in [0 , n -dimensional generalization isas follows: ( H f )( x ) = Z R m K ( u ) f ( A ( u ) x ) du, (1)where K : R m → C is a locally integrable function, f : R n → C , A ( u ) standsfor a family of non-singular n × n -matrices with real entries defined on R m , x ∈ R n is a column vector [2]. The modern theory of Hausdorff operators wasinitiated by Liflyand and Moricz [3]. See survey articles [4], [5] for historicalremarks and the state of the art up to 2014.Note that the map x Ax ( A ∈ GL ( n, R )) which appears in the formula(1) is the general form of an automorphism of the additive group R n . This1bservation leads to the definition of a (generalized) Hausdorff operator over ageneral group G via the automorphisms of G that was introduced and studiedby the author in [6], and [7] . For the additive group R n , this definition looksas follows: Definition 1.
Let (Ω , µ ) be a σ -compact topological space endowed witha positive regular Borel measure µ, let K be a locally integrable function onΩ , and let ( A ( u )) u ∈ Ω be a µ -measurable family of n × n -matrices, non-singularfor µ -almost every u , with K ( u ) = 0 . We define the
Hausdorff operator withthe kernel K by (recall that x ∈ R n is a column vector)( H K,A f )( x ) = Z Ω K ( u ) f ( A ( u ) x ) dµ ( u ) . The general form of a Hausdorff operator given by Definition 1 (withan arbitrary measure space (Ω , µ ) instead of R m ) gives us, for example, theopportunity to consider (in the case Ω = Z m ) discrete Hausdorff operators(see [12], [13], and Example 3 below).The problem of compactness of Hausdorff operators was posed by Liflyand[14] (see also [4]). There is a conjecture that non-zero Hausdorff operatoron L p ( R n ) is non-compact. For the case p = 2 and for commuting A ( u )this hypothesis was confirmed in [12] (and for the diagonal A ( u ) — in [6]).Moreover, we conjectured in [15] that a nontrivial Hausdorff operator on L p ( R n ) is non-Riesz.A notion of a Riesz operator was introduced by Ruston [16]. Recall that abounded operator T on some Banach space is a Riesz operator if it possessesspectral properties like those of a compact operator; i. e., T is a non-invertibleoperator whose nonzero spectrum consists of eigenvalues of finite multiplicitywith no limit points other then 0. This is equivalent to the fact that T − λ is Fredholm for all scalars λ = 0 [17], [18, Section 9.6]. For example, a sumof a quasinilpotent and compact operator is Riesz [19, Theorem 3.29]. See[17], [18, Section 9.6], [19] and the bibliography therein for other interestingcharacterizations of Riesz operators.In this note we prove the aforementioned conjecture for the case where A ( u ) is a commuting family of self-adjoint matrices. The result has beenannounced in [20]. The case of positive or negative definite perturbationmatrices was considered in [15]. The case of a Hausdorff operator on p -Adic vector spaces was considered earlier in[8], the special case of a Hausdorff operator on the Heisenberg group in the sense of thisdefinition was considered in [9], and [10]. The generalized Delsart translation operators(see, e.g., [11]) lead to Hausdorff operators in the sense of this definition, too The main result
We shall employ for lemmas to prove our main result.
Lemma 1 [6] (cf. [1, (11.18.4)], [21]).
Let | det A ( u ) | − /p K ( u ) ∈ L (Ω) . Then the operator H K,A is bounded in L p ( R n ) ( ≤ p ≤ ∞ ) and kH K,A k ≤ Z Ω | K ( u ) || det A ( u ) | − /p dµ ( u ) . This estimate is sharp (see Theorem 1 in [13]).
Lemma 2 [13] (cf. [21]).
Under the assumptions of Lemma the adjointfor the Hausdorff operator on L p ( R n ) is of the form ( H ∗ K,A f )( x ) = Z Ω K ( v ) | det A ( v ) | − f ( A ( v ) − x ) dµ ( v ) . Thus, the adjoint for a Hausdorff operator is also Hausdorff.
Lemma 3.
Let S be a ball in R n , q ∈ [1 , ∞ ) , and let R q,S denote therestriction operator L q ( R n ) → L q ( S ) , f f | S . If we identify the dual ( L q ) ∗ of L q with L p ( /p + 1 /q = 1 ), then the adjoint R ∗ q,S is the operator of naturalembedding L p ( S ) ֒ → L p ( R n ). Proof.
For g ∈ L p ( S ), let g ∗ ( x ) = ( g ( x ) , for x ∈ S, , for x ∈ R n \ S. Then the map g g ∗ is the natural embedding L p ( S ) ֒ → L p ( R n ).By definition, the adjoint R ∗ q,S : L q ( S ) ∗ → L q ( R n ) ∗ acts according to therule ( R ∗ q,S Λ)( f ) = Λ( R q,S f ) (Λ ∈ L q ( S ) ∗ , f ∈ L q ( R n )) . If we (by the Riesz theorem) identify the dual of L q ( S ) with L p ( S ) via theformula Λ ↔ g , whereΛ( h ) = Z S g ( x ) h ( x ) dx ( g ∈ L p ( S ) , h ∈ L q ( S )) , and analogously for the dual of L q ( R n ), then the definition of R ∗ q,S is of theform Z R n ( R ∗ q,S g )( x ) f ( x ) dx = Z S g ( x )( f | S )( x ) dx. But Z S g ( x )( f | S )( x ) dx = Z R n g ∗ ( x ) f ( x ) dx ( f ∈ L q ( R n )) . L q ( R n ) ∗ .If we (again by the Riesz theorem) identify this functional with the function g ∗ , the result follows.Consider the modified n -dimensional Mellin transform for the n -hyperoctant U of R n in the form( M f )( s ) := 1(2 π ) n/ Z U | x | − q + is f ( x ) dx, s ∈ R n . Here and below we assume that 1 < q ≤ ∞ , x | − q + is := n Y j =1 | x j | − q + is j , where | x j | − q + is j := exp (cid:18)(cid:18) − q + is j (cid:19) log | x j | (cid:19) . Lemma 4. The map M is a bounded operator between L p ( U ) and L q ( R n ) for ≤ p ≤ ( /p + 1 /q = 1) . If we identify the dual ( L p ) ∗ of L p with L q ( /p + 1 /q = 1 ), then theadjoint for the operator M on the space L p ( U ) ( ≤ p ≤ ) is as follows: ( M ∗ g )( x ) = 1(2 π ) n/ Z R n | x | − q + is g ( s ) ds, x ∈ U. Proof. 1) It can easily be obtained from the Hausdorff–Young inequalityfor the n -dimensional Fourier transform by using the exponential change ofvariables (see [22]).2) To compute M ∗ , for g ∈ L p ( R n ) consider the operator( M ′ g )( x ) := 1(2 π ) n/ Z R n | x | − q + is g ( s ) ds, x ∈ U. This is a bounded operator taking L p ( R n ) into L q ( U ) . Indeed, since | x | − q + is = n Y j =1 | x j | − q exp( is j log | x j | ) , we have( M ′ g )( x ) = | x | − q π ) n/ Z R n exp( is · (log | x j | )) g ( s ) ds, x ∈ U, | x | := | x | . . . | x n | , (log | x j | ) := (log | x | , . . . , log | x n | ), and the dot de-notes the inner product on R n . Thus, we can express the function M ′ g viathe Fourier transform b g of g as( M ′ g )( x ) = | x | − /q b g ( − (log | x j | )) , x ∈ U, and therefore kM ′ g k L q ( U ) = (cid:18)Z U | x | − | b g ( − (log | x j | )) | q dx (cid:19) /q . Putting in the last integral y j := − log | x j | ( j = 1 , . . . , n ) and taking intoaccount that the Jacobian module of this transformation is (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( x , . . . , x n ) ∂ ( y , . . . , y n ) (cid:12)(cid:12)(cid:12)(cid:12) = det diag( e − y , . . . , e − y n ) = exp − n X j =1 y j ! , we get by the Hausdorff–Young inequality that kM ′ g k L q ( U ) = k b g k L q ( R n ) ≤ k g k L p ( R n ) . If f ∈ L p ( U ), and f ( x ) | x | − /q ∈ L ( U ), g ∈ L p ( R n ) ∩ L ( R n ), then theFubini–Tonelli’s theorem implies Z R n ( M f )( s ) g ( s ) ds = Z U f ( x )( M ′ g )( x ) dx. Since the bilinear dual pairing ( ϕ, ψ ) R ϕψdν is continuous on L p ( ν ) × L q ( ν ), the last equality is valid for all f ∈ L p ( U ), g ∈ L p ( R n ) by continuity.So, M ′ = M ∗ .Now we are in position to prove our main result. Theorem 1.
Let A ( u ) be a commuting family of real self-adjoint n × n -matrices ( u satisfies K ( u ) = 0 ), and (det A ( u )) − /p K ( u ) ∈ L (Ω) . Thena Hausdorff operator H K,A in L p ( R n ) ( ≤ p ≤ ∞ ) is non-Riesz (and inparticular it is not a sum of a quasinilpotent and compact operator) providedit is non-zero. Proof. Assume the contrary. Since A ( u ) form a commuting family, thereare an orthogonal n × n -matrix B and a family of diagonal non-singular realmatrices A ′ ( u ) = diag( a ( u ) , . . . , a n ( u )) such that A ′ ( u ) = B − A ( u ) B for u ∈ Ω . Consider the bounded and invertible operator ( b Bf )( x ) := f ( Bx ) on L p ( R n ) . Because of the equality b B H K,A b B − = H K,A ′ , H K,A ′ is Riesz and nontrivial, too. For the proof of the Riesz prop-erty it is sufficient to verify the Ruston condition [16] (see also [19, Theorem3.12]) for this operator. But if C denotes the ideal of compact operators on L p ( R n ) one haveinf C ′ ∈C kH nK,A ′ − C ′ k /n = inf C ′ ∈C k b B H nK,A b B − − C ′ k /n ≤ k b B k /n inf C ′ ∈C kH nK,A − b B − C ′ b B k /n k b B k − /n = k b B k /n inf C ∈C kH nK,A − C k /n k b B k − /n → n → ∞ which proves the Ruston condition for H K,A ′ .As in [12] let’s consider some fixed enumeration U j ( j = 1; . . . ; 2 n ) of thefamily of all open hyperoctants in R n . For every pair ( i ; j ) of indices there isa unique sequence ε ( i ; j ) ∈ {− , } n such that ε ( i ; j ) U i := { ( ε ( i ; j ) x ; . . . ; ε ( i ; j ) n x n ) : x = ( x k ) nk =1 ∈ U i } = U j . Then ε ( i ; j ) U j = U i and ε ( i ; j ) U l ∩ U i = ∅ as l = j. We put Ω ij := { u ∈ Ω : (sgn( a ( u )); . . . ; sgn( a n ( u ))) = ε ( i ; j ) } and let ( H ij f )( x ) := Z Ω ij K ( u ) f ( A ′ ( u ) x ) dµ ( u ) . Since f ( A ′ ( u ) x ) = 0 for f ∈ L p ( U j ) and x / ∈ U i , each H ij maps L p ( U j ) into L p ( U i ). Moreover, if f ∈ L p ( R n ) and f j := f χ U j ( χ E denotes the indicatorof a subset E ⊂ R n ) then as in the proof of formula (1) in [12] it is easy toverify that for a. e. x ∈ R n ( H K,A ′ f )( x ) = n X j =1 2 n X i =1 ( H ij f j )( x ) . It follows that for some pare ( i, j ) of indices the restriction H := H ij | L p ( U j )is a nontrivial operator. In the sequel the pare ( i, j ) will be fixed.Consider the map J x := ( ε ( i, j ) k x k ) nk =1 ( x = ( x k ) nk =1 ∈ R n ) . Then J : U j → U i and the operator ( b J f )( x ) := f ( J x ) maps L p ( U i ) on L p ( U j )isometrically. It follows that the operator K := b J H L p ( U j ) and is bounded. Hereafter we put U := U j for simplicity. Then K is a nontrivial Riesz operator on L p ( U ). Indeed, the operator b J H K,A ′ = H K,A ′ b J is Riesz on L p ( R n ) [17, Lemma 5]. Since the space L p ( U ) is invariantwith respect to this operator, the restriction b J H K,A ′ | L p ( U ) (which is equalto K ) is Riesz on L p ( U ) by [19, p. 80, Theorem 3.21], as well.Let 1 ≤ p < ∞ . To get a contradiction, we shall use the modified n -dimensional Mellin transform M for the n -hyperoctant U . By Lemma 4 themap M is a bounded operator between L p ( U ) and L q ( R n ) for 1 ≤ p ≤ /p + 1 /q = 1) . Let f ∈ L p ( U ). Note that( K f )( x ) = Z Ω ij K ( u ) f ( A ′ ( u )( J x )) dµ ( u ) = Z Ω ij K ( u ) f ( A ′′ ( u ) x ) dµ ( u )where A ′′ ( u ) = diag( ε ( i, j ) a ( u ) , . . . , ε ( i, j ) n a n ( u )) . First assume that | y | − /q f ( y ) ∈ L ( U ) . Then, as in the proof of Theorem 1from [13] (or [12]), using Fubini–Tonelli’s theorem and integrating by substi-tution x = ( A ( u ) ′′ ) − y yield( MK f )( s ) = ϕ ( s )( M f )( s ) ( s ∈ R n ) , where the function ϕ ( s ) := Z Ω ij K ( u ) | a ( u ) | − /p − is dµ ( u )(the ( i ; j ) entry of the matrix symbol of a Hausdorff operator [13, Definition2], [12]) is bounded and continuous on R n . Thus, MK f = ϕ M f. (2)By continuity, the last equality is valid for all f ∈ L p ( U ) . Let 1 ≤ p ≤ . There exists a constant c > { s ∈ R n : | ϕ ( s ) | > c } contains an open ball S. Formula (2) implies that M ψ R q,S MK = R q,S M , where ψ = (1 /ϕ ) | S, M ψ denotes the operator of multiplication by ψ , and R q,S : L q ( R n ) → L q ( S ) , f f | S is the restriction operator. Let T = R q,S M . Passing to the conjugates gives K ∗ T ∗ M ∗ ψ = T ∗ .
7y [23, Theorem 1] this implies that the operator T ∗ = M ∗ R ∗ q,S has finiterank. But by Lemma 3, R ∗ q,S is the operator of natural embedding L p ( S ) ֒ → L p ( R n ). Thus, the restriction of the operator M ∗ to L p ( S ) has finite rank.Since by Lemma 4 M ∗ can easily be reduced to the Fourier transform, thiscontradicts the Paley–Wiener theorem on the Fourier image of the space L ( S ), see, e. g., [24, Theorem III.4.9] (in our case L ( S ) ⊂ L p ( S )).Finally, if 2 < p ≤ ∞ , one can use duality arguments. Indeed, by Lemma2 the adjoint operator H ∗ K,A ′ (as an operator on L q ( R n )) is also of Hausdorfftype. More precisely, it equals H Ψ ,B , where B ( u ) = A ( u ) ′− = diag(1 /a ( u ) , . . . , /a n ( u ))and Ψ( u ) = K ( u ) | det A ( u ) ′− | = K ( u ) / Y j a j ( u ) . It is easy to verify that H Ψ ,B satisfies all the conditions of Theorem 1 (with q, Ψ and B in place of p, K and A , respectively). Since 1 ≤ q <
2, theoperator H Ψ ,B is not a Riesz operator on L q ( R n ). The same is true for H K,A ,because T is a Riesz operator if only if its conjugate T ∗ is a Riesz operator[19, p. 81, Theorem 3.22]. This completes the proof. Corollary 1 . Under the same assumptions of Theorem 1 the operator H K,A − λ is not Fredholm for some scalar λ = 0 . Corollary 2
Let the conditions of theorem 1 hold. Then either there isa non-zero point of σ ( H K,A ) that is not a pole of the resolvent of H K,A , orthere is a non-zero point λ of σ ( H K,A ) such that the spectral projection E ( λ ) has infinite-dimensional range. Proof. This follows from Theorem 1 and the characterization of Rieszoperator given in [19, Theorem 3.17].
Corollary 3 [15].
Let A ( v ) be a commuting family of real positive or nega-tive definite n × n -matrices ( v runs over the support of K ), and det A ( v ) − p K ( v ) ∈ L (Ω) . Then a Hausdorff operator H K,A in L p ( R n ) ( ≤ p ≤ ∞ ) is non-Riesz (and, in particular, it is not the sum of a quasinilpotent and compactoperators) provided it is non-zero. Corollary 4.
Let φ : Ω → C , and a : Ω → R be measurable func-tions, such that | a ( u ) | − /p φ ( u ) ∈ L (Ω) . Then a one-dimensional Hausdorff perator ( H φ,a f )( x ) = Z Ω φ ( u ) f ( a ( u ) x ) dµ ( u ) ( x ∈ R ) on L p ( R ) ( ≤ p ≤ ∞ ) is non-Riesz (and, in particular, it is not the sum ofa quasinilpotent and compact operators) provided it is non-zero. Example 1 . Let t − /q ψ ( t ) ∈ L (0 , ∞ ) . Then, by Corollary 4, the opera-tor ( H ψ f )( x ) = Z ∞ ψ ( t ) t f (cid:16) xt (cid:17) dt is a non-Riesz operator in L p ( R ) (1 ≤ p ≤ ∞ ) provided it is non-zero. Example 2 . Let ( t t ) − /p ψ ( t , t ) ∈ L ( R ) . Then, by Corollary 3, theoperator ( H ψ f )( x , x ) = 1 x x Z ∞ Z ∞ ψ (cid:18) t x , t x (cid:19) f ( t , t ) dt dt is a non-Riesz operator in L p ( R ) (1 ≤ p ≤ ∞ ) provided it is non-zero. Example 3 . (Discrete Hausdorff operators, cf. [12, Example 3]). LetΩ = Z m + , and µ be a counting measure. Then Definition 1 turns into ( f ∈ L p ( R n ), 1 ≤ p ≤ ∞ ) ( H K,A f )( x ) = X u ∈ Z m + K ( u ) f ( A ( u ) x )( A ( u ) form a family of real non-singular n × n matrices). Assume that P u ∈ Z m + | K ( u ) || det A ( u ) | − /p < ∞ . Then the operator H K,A is well definedand bounded on L p ( R n ) by Lemma 1, and is a non-Riesz operator by Theorem1 provided it is non-zero and matrices A ( u ) are permutable and self-adjoint. Acknowledgment.
This work was supported by the State Program ofScientific Research of Republic of Belarus.This is a preprint of the article [25].
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