Weighted Composite Integral Operators

Int. Journal of Math. Analysis, Vol. 3, 2009, no. 26, 1283 - 1293

Weighted Composite Integral Operators Anupama Gupta G.D.C. Akhnoor, Jammu, India [email protected] B. S. Komal Department of Mathematics University of Jammu, Jammu, India [email protected] Abstract In this paper we explore the basic properties of weighted composite integral operators and weighted composite operators of Hardy type. We also obtain the spectra of weighted composite operators of Hardy type.

Mathematics subject classification: Primary 47B20, Secondary: 46B38 Keywords: Composition operator, Integral operator, Hardy operator, Hermitian operator, Idempotent operator

INTRODUCTION In recent years a great deal of work was done on composition operators as well as on integral operators. The adjoints of the unilateral shift, the bilateral shift and the weighted shift on 2 (Z) are well studied examples of composition operators. For more detail about composition operators we refer to Nordgen [4], Shapiro [8], Cowen [3], Singh and Manhas [12], Singh and Komal [11]. The integral operators received considerable attention of several mathematicians like Stepanov ([15],[16]), Sinnamon [5], Halmos and Sunder [10], Bloom and Kerman [14], Gupta and Komal [1]. In his paper Whitley [13] established the Lyubic’s conjecture [6] and generalized it to Volterra composition operators on Lp [0, 1]. The theory of Hardy operator is a source of modern functional analysis and operator theory. Weighted norm inequalities for operators of Hardy type are obtained by mathematicians like Sinnamon [5], Stepanov [15], Bloom and Kerman [14] etc. Let (X, S, μ) be a σ-finite measure space and let φ : X → X be a non-singular

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A. Gupta and B. S. Komal

measurable transformation (μ(E) = 0 ⇒ μφ−1 (E) = 0). Then a composition transformation, for 1 ≤ p < ∞, Cφ : Lp (μ) → Lp (μ) is defined by Cφ f = f oφ for every f ∈ Lp (μ). In case Cφ is continuous, we call it a composition operator induced by φ. It is easy to see that Cφ is a bounded operator if and only if dμφ−1 = f0 , the Radon-Nikodym derivative of the measure μφ−1 with respect dμ to the measure μ, is essentially bounded. A kernel K ∈ Lp (μ × μ) always induces a bounded integral operator TK : Lp (μ) → Lp (μ) defined by

(TK f )(x) =

K(x, y)f (y)dμ(y).

Suppose u : X → C is a measurable function. Then the bounded operator Wu,φ,K defined by (Wu,φ,K f )(x) = u(x)K(φ(x), y)f (y)dμ(y) is known as weighted composite integral operator. Set Kφ (x, y) = (E(K(x, .))oφ−1 fo )(y) = E(K(x, φ−1 (y))fo (y)) = E(Kx (φ−1 (y))fo(y)) where E is called conditional expectation. For more properties of the expectation operator, see Parthasarthy [9], and Lambert [2]. The Hardy operator H : L1 [0, 1] → L1 [0, 1] is defined by x f (y)dμ(y) x > 0. (Hf )(x) = 0

If φ : [0, 1] → [0, 1] is a measurable transformation, then the composite Hardy operator is defined by φ(x) f (y)dμ(y). (Hφ f )(x) = (Hf )oφ(x) = 0

Let u : [0, 1] → C be a measurable function. The weighted composite operator of Hardy type Hφ,u : L1 [0, 1] → L1 [0, 1] is defined by

(Hφ,u f )(x) = u(φ(x))

φ(x)

f (y)dμ(y). 0

Weighted composite integral operators

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Our main purpose in this paper is to characterize the basic properties of more general operators called weighted composite integral operators and weighted composite operators of Hardy type. It is often very difficult to compute the adjoint of a composition operator on L2 (μ) which for many years remained unknown to us. In 1990, Campbell and Jamison [7] obtained the adjoint of a composition operator on L2 (μ). We also explore the adjoint of a weighted composite integral operator. A condition for a weighted composite integral operator to be Hermitian is given. The weighted composite operators of Hardy type are studied. It is shown that the spectrum of a weighted composite operator of Hardy type is consisting of 0 only. The symbol B(L2 (μ)) denotes the Banach algebra of all bounded linear oper2 2 2 ators from L (μ) to L (μ) and K (x, y) is defined by K(x, z)K (z, y)dz. 1. Hermitian Weighted Composite Integral Operators In this section we first give a sufficient condition for a weighted composite integral operator to be bounded. Theorem 1.1: Let Gu,φ ∈ L2 (μ × μ). Then Wu,φ,K is bounded weighted composite integral operator, where Gu,φ (x, y) = fo (x)E(|u|2oφ−1 )(x)K(x, y). Proof: For every f ∈ L2 (μ), we have

||Wu,φ,K f ||

2

= = ≤ ≤ =

|(Wu,φ,K f )(x)|2 dμ(x) X | u(x)K(φ(x), y)f (y)dμ(y)|2dμ(x) X 2 |u(x)K(φ(x), y)| dμ(y)dμ(x) |f (y)|2dμ(y) ( fo (x)E(|u|2 oφ−1 )(x)|K(x, y)|2dμ(x)dμ(y))(||f ||2) X X 2 ||f || ( |Gu,φ (x, y)|2dμ(x)dμ(y)) X

X

= ||Gu,φ ||2 .||f ||2.

Hence Wu,φ,K is a bounded operator.

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In the following theorem we compute the adjoint of a weighted composite integral operator. Let A : L2 (μ) → L2 (μ) be defined by (Af )(x) = fo (y)K(y, x)E(uf )oφ−1(y)dμ(y). ∗ . Theorem 1.2: If Wu,φ,K ∈ B(L2 (μ)), then A = Wu,φ,K Proof : Consider

< f, Wu,φ,K g > =

f (x)(Wu,φ,K g)(x)dμ(x) = f (x) u(x)K(φ(x), y)g(y)dμ(y)dμ(x). = E((uf )oφ−1(x))fo (x)K(x, y)dμ(x)g(y)dμ(y) = (Af )(y)g(y)dμ(y).

= < Af, g > for every f, g ∈ L2 (μ). ∗ . Hence A = Wu,φ,K Theorem 1.3: Let Wu,φ,K ∈ B(L2 (μ)) and φ−1 (S) = S. Then Wu,φ,K is Hermitian if and only if fo (x)K(x, y)E(uoφ−1 )(x)dμ(x)dμ(y) E

φ−1 (F )

= φ−1 (E)

fo (y)K(y, x)E(uoφ−1)(y)dμ(y)dμ(x).

F

Proof: Suppose Wu,φ,K is Hermitian. Then ∗ ∗ < Wu,φ,K χφ−1 (E) , χφ−1 (F ) >= Wu,φ,K χφ−1 (E (x)χφ−1 (F ) (x)dμ(x) = fo (y)K(y, x)E(uχφ−1 (E) oφ−1 (y))dμ(y)χφ−1(F dμ(x) = fo (y)K(y, x)E(uχE oφ−1 )(y))dμ(y)χφ−1(F ) dμ(x) = fo (y)K(y, x)E(uoφ−1)(y)dμ(y)dμ(x) ...(1) E

φ−1 (F )


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and < Wu,φ,K χφ−1 (E) , χφ−1 (F ) >= Wu,φ,K χφ−1 (E) (y)χφ−1(F ) (y)dμ(y) = u(y)K(φ(y), x)χφ−1(E) dμ(x)χφ−1 (F ) dμ(y) = fo (y)K(y, x)E([u]oφ−1)(y))χF (y)dμ(y)χφ−1(E (x)dμ(x) fo (y)K(y, x)E([u]oφ−1)(y)dμ(y)dμ(x) ...(2) = φ−1 (E)

F

From (1) and (2), it is clear that if Wu,φ,K is Hermitian, then the condition must hold. Conversely, if the condition is true, then clearly ∗ < Wu,φ,K χφ−1 (E) , g >=< Wu,φ,K χφ−1 (E) , g >

for every simple function g. Since simple functions are dense in Lp (u), so ∗ χφ−1 (E) , g >=< Wu,φ,K χφ−1 (E) , g > < Wu,φ,K

for every g ∈ L2 (μ). This proves that Wu,φ,K is Hermitian. Theorem 1.4: Let Wu,φ,K ∈ B(L2 (μ)). Then Wu,φ,K is idempotent if and only if Kφ is idempotent and |u| = 1 a.e. Proof: If the conditions are true, then an easy computation shows that Wu,φ,K is idempotent Consider 2 ∗ < Wu,φ,K f, g > = < Wu,φ,K f, Wu,φ,K g> ∗ = Wu,φ,K f (x).Wu,φ,K g(x)dμ(x) X = ( u(x)K(φ(x), y)f (y)dμ(y)) X u(x)K(φ(x), z)g(z)dμ(z)dμ(x) . |u(x)|2 K(φ(x), y)f (y)K(z, φ(x)) = X

. =

g(z)dμ(z)dμ(y)dμ(x) Wu,φ,K f (x)g(x)dμ(x)

= < Wu,φ,K f, g) >

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Hence Wu,φ,K is an indempotent operator. Conversely, if Wu,φ,K is an idempotent operator, then taking f = χE , g = χF . 2 χE , χF >=< Wu,φ,K χE , χF > < Wu,φ,K

2 |u(x)| .K(z, φ(x))K(φ(x), y)dμ(x)f (y)g(z)dμ(y)dμ(z) = ( u(x)K(z, y)f (y)dμ(y))g(z)dμ(z) Thus 2 2 |u(x)| K (φ(z), y)dμ(y) = u(x)K(φ(z), y)dμ(y)

i.e.

E×F

E×F 2

which proves that |u| = 1 a.e. and K (φ(z), y) = K(φ(z), y). 2. Weighted Composite Operators of Hardy type In this section we first obtain criterion for a bounded weighted composite operators of Hardy type on L1 (μ) as well as on L2 (μ). Theorem 2.1: Let Hφ,u : L1 (μ) → L1 (μ) be a mapping. Then Hφ,u is a bounded operator if and only if there exists M > 0 such that H ∗ (|u|fo) ≤ M. Proof: Suppose the condition is true. Consider ||Hφ,u f || =

∞

0 ∞

= 0 ∞

|(Hφ,u f )(x)|dμ(x) |u(φ(x))(Hf )(φ(x))|dμ(x)

f0 |u(x)|.|Hf (x)|dμ(x) x ∞ f0 (x)u(x)( |f (y)|dμ(y))dμ(x) = 0 0 ∞ x = f0 (x)u(x)|f (y)|dμ(y)dμ(x)

≤

0

0

0

(by interchanging order of integration)

∞

|f (y)|(

=

0

≤ M

fo (x)u(x)dμ(x))dμ(y) y

∞

= 0

∞

|f (y)|H ∗(fo |u|)(y)dμ(y)

0

∞

|f (y)|dμ(y) = M||f ||.


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Thus Hφ,u is a bounded weighted composite operator of Hardy type. Conversely, suppose H ∗ (ufo ) is not a bounded function. Then for every positive integer n, there exists a measurable subset En , 0 < μ(En ) < 1 such that H ∗ (ufo )(x) ≥ n for μ almost all x ∈ En . χ Take fn = μ(EEnn ) . Then ||fn || = 1. and

||Hφ,u fn || = = = = ≥

∞

|Hφ,u fn (x)|dμ(x) x ∞ 1 fo (x)|u(x)|( χEn (y)dμ(y))dμ(x) μ(En ) 0 0 ∞ ∞ 1 ( fo (x)|u(x)|dμ(x)(χEn (y)dμ(y)) μ(En ) 0 y ∞ 1 H ∗ (fo |u|)(y)χEn (y)dμ(y) μ(En ) 0 ∞ n χEn (y)dμ(y) = n||fn || μ(En ) 0 0

which contradicts the fact that Hφ,u is a bounded operator. Hence H ∗ (ufo ) must be a bounded function. Theorem 2.2: Let Hφ,u : L2 (μ) → L2 (μ) be a mapping. Then Hφ,u is bounded if and only if there exists M > 0 such that

H(fo |u|2 ) ≤ M

Proof: For f ∈ L2 (μ)

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||Hφ,uf ||

2

=

|Hφ,u f (x)|2 dx ∞

= x=0 ∞

=

|u(φ(x))(Hf )(φ(x))|2dx

fo |u(x)|2|Hf (x)|2 dx x 2 fo |u(x)| ( |f (y)|dy)2dx 0 x x x=0 ∞ 2 fo |u(x)| ( |f (y)|dy. |f (z)|dz)dx x=0 0 0 x ∞ 2 fo |u(x)| ( |f (y)|dys(x))dx x=0 0 ∞ ∞ |f (y)|dy fo (x)|u(x)|2 s(x)dx x=y x y=0 ∞ ∞ 2 |f (y)|dy fo (x)|u(x)| ( |f (z)|dz)dx y=0 x=y 0 ∞ z ∞ |f (y)|dy f (z)[ fo (x)|u(x)|2 dx]dz x=0 z=0 y=0 ∞ ∞ |f (y)|dy. f (z)(H(fo |u|2))(z)dz 0 0 ∞ ∞ |f (y)|dy. f (z)dz M 0 0 ∞ |f (y)|2dy = M.||f ||2 . M x=0 ∞

= = = = = = = ≤ =

0

Conversely, suppose H(fo |u|2) is not bounded. Then for every positive integer n, there exists a measurable subset En , 0 < μ(En ) < 1 such that H(fo (x)|u(x)|2 ) ≥ n for μ almost all x ∈ En . χ n Take fn = [μ(EnE)]1/2 , ||fn || = 1.


||Hφ,u fn ||

2

∞

= x=0 ∞

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|u(φ(x))(Hf )(φ(x))|2dx

fo |u(x)|2.|Hfn (x)|2 dx x 0 ∞ 2 fo |u(x)| .| fn (y)dy|2dx 0 ∞ 0∞ x 2 |fn (y)|dy fo (x)|u(x)| . |fn (z)dz. 0 y 0 ∞ ∞ |fn (y)|dy. |fn (z)H(fo (|u|2 )(z)dz. 0 0 ∞ n |χEn (y)|2d(y) = n||fn ||2. μ(En ) 0

≤ = = = ≥

In the next result an attempt has been made to explore the adjoint of weighted composite operator of Hardy type. Theorem 2.3: Let Hφ,u ∈ B(L1 (μ)). Then ∗ Hφ,u f = H ∗ (ufo .f )

Proof: For g ∈ L1 (μ) and f ∈ L∞ (μ). Consider ∗ < Hφ,u f, g > = < f, Hφ,u g > ∞ f (x)Hφ,u g(x)dμ(x) = 0 φ(x) ∞ f (x)(u(φ(x)) g(y)dμ(y))dμ(x) = 0 0 ∞ x = f (x)( u(x)fo (x)g(y)dμ(y))dμ(x) 0 0 ∞ ∞ ( fo (x)u(x)f (x)dμ(x))g(y)dμ(y) = 0 y ∞ (H ∗ (ufo f ))(y)g(y)dμ(y) = 0

= < H ∗ (ufo f ), g > ∗ = H ∗ (ufo f ). Hence Hφ,u

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In the next result it is proved that spectrum of weighted composite operator of Hardy type is equal to zero. Theorem 2.4: Suppose φ(0) = 0. Then σ(Hφ,u ) = {0}. Proof: If possible, suppose σ(Hφ,u ) = {0}. Suppose λ = 0, λ is an eigen value of Hφ,u . Hφ,u f (x) = λf (x)

...(1)

Also u(φ(x)Hf (φ(x)) = λf (x)

φ(x)

u(φ(x))f (y)dy = λf (x) y=0

Differentiate with respect to x.

φ(x)

u(φ(x))f (φ(x))φ (x) + u (φ(x))φ (x).

f (y)dy = λf (x)

...(2)

0

From (1), we have f(0) = 0. From (2), we have f (0) = 0. Also f (0) = 0,...... and so on f n (0) = 0. Thus f = 0, which is a contradiction. Hence σ(Hφ,u ) = {0}. REFERENCES 1. A. Gupta and B.S. Komal, Composite integral operator on L2 (μ), Pitman Lecture Notes in Mathematics series 377, (1997), 92-99. 2. A. Lambert, Normal extension of subnormal composition operators, Michigan Math. J., 35, (1988), 443-450. 3. C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced mathematics CRC Press, New York 1995. 4. E.A. Nordgen, Composition operators on Hilbert spaces, Lectures notes in Math, 693, Springer Verlag, New York, (1978), 37-63. 5. G. Sinnamon, Weighted Hardy and opial type inequalities J. Math. Anal. Appl. 160, (1991), 434-435 6. I. Yu. Lyubic, Composition of integration and substitution, Linear and complex Analysis, Problem Book, Springer Lect. Notes in Maths., 1043, Berlin (1984), 249-250


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7. J.Campbell and J. Jamison, On some classes of weighted composition operators, Glasgow Math. J. 32, (1990), 87-94 8. J. H. Shapiro, Composition operators and classical function theory, SpringerVerley, New York. 1993. 9. K. R. Parthasarathy, Introduction to probability and measure, Macmillion Limited, 1977. 10. P. R. Halmos and V.S. Sunder, Bounded integral operators on L2 -spaces, Springer-Verlag, New York, 1978. 11. R. K. Singh and B. S. Komal, Composition operator on I p and its adjoint, Proc. Amer. Math. Soc. 70, (1978), 21-25. 12. R.K. Singh and J. S. Manhas, Composition operators on function spaces, North Holland Mathematics studies 179, Elsevier sciences publishers Amesterdam, New York 1993. 13. R. Whitley, The spectrum of a Volterra composition operator, Integral equation and operator theory Vol. 10, 1997. 14. S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113, (1991), 135-141. 15. V. D. Stepanov, Weighted norm inequalities of Hardy type for a class of integral operators, J. London Math. Soc. (2) 50, (1994), 105-120. 16. V. D. Stepanov, Weighted inequalities for a class of Volterra convolution operators, J. London Math. (1992), 232-242 Received: January, 2009