Operators satisfying the condition [[parallel][T.sup.2+k]x[parallel].sup.1/1+k] [[parallel]Tx[parallel].sup.k1+k] [greater than or equal to] [parallel][T.sup.2]x[parallel].

[section] 1. Introduction

Let B(H) be the Banach Algebra of all bounded linear operators on a non-zero complex Hilbert space H. By an operator, we mean an element from B(H). If T lies in B(H), then [T.sup.*] denotes the adjoint of T in B(H). For 0 < p [less than or equal to] 1, an operator T is said to be p-hyponormal if [([T.sup.*]T).sup.p] [greater than or equal to] [(T[T.sup.*]).sup.p]. If p = 1, T is called hyponormal. If p = 1/2, T is called semi-hyponormal. An operator T is called paranormal, if [[parallel][T.sup.x][parallel].sup.2] [less than or equal to] [parallel][T.sup.2]x[parallel] [parallel]x[parallel], for every x [member of] H. An operator T is normaloid if r(T) = [parallel]T[parallel], where r(T) is the spectral radius of T or [[parallel]T[parallel].sup.n] = [parallel][T.sup.n][parallel] for all positive integers n.

In general, hyponormal [??] p-hyponormal [??] paranormal [??] k-paranormal.

Ando [4] has characterized paranormal operators as follows:

Theorem 1.1. An operator T [member of] B(H) is paranormal if and only if [T.sup.*2][T.sup.2]-2k[T.sup.*]T+[k.sup.2] [greater than or equal to] 0, for every k [member of] R.

Generalising this, Yuan and Gao [13] has characterised k-paranormal operators as follows:

Theorem 1.2. For each positive integer k, an operator T [member of] B(H) is k-paranormal if and only if [T.sup.*1+k][T.sup.1+k] - (1 + k)[[mu].sup.k][T.sup.*]T + k[[mu].sup.1+k]I [greater than or equal to] 0, for every [mu] > 0.

In [10], Uchiyama gives a matrix representation for a paranormal operator with respect to the direct sum of an eigenspace and its orthogonal complement.

In this paper we characterize the new class of operators which properly contains k-paranorm -al operators, discuss its matrix representation and prove some more properties. We also characterize the composition operators and weighted composition operators of this class.

[section] 2. Preliminaries

Let (X, [SIGMA], [lambda]) be a sigma-finite measure space. The relation of being almost everywhere, denoted by a.e, is an equivalence relation in [L.sup.2](X, [SIGMA], [lambda]) and this equivalence relation splits [L.sup.2](X, [SIGMA], [lambda]) into equivalence classes. Let T be a measurable transformation from X into itself. [L.sup.2](X, [SIGMA], [lambda]) is denoted as [L.sup.2]([lambda]). The equation [C.sub.T]f = f [omicron] T, f [member of] [L.sup.2]([lambda]) defines a composition transformation on [L.sup.2]([lambda]). T induces a composition operator [C.sub.T] on [L.sup.2]([lambda]) if (i) the measure [lambda][omicron][T.sup.-1] is absolutely continuous with respect to [lambda] and (ii) the Radon-Nikodym derivative d([lambda][T.sup.-1]/d[lambda] is essentially bounded (Nordgren). Harrington and Whitley have shown that if [C.sub.T] [member of] B([L.sup.2]([lambda])), then [C.sup.*.sub.T][C.sub.T]f = [f.sub.0]f and [C.sub.T][C.sup.*.sub.T]f = ([f.sub.0] [omicron] T)P f for all f [member of] [L.sup.2]([lambda]) where P denotes the projection of [L.sup.2]([lambda]) onto [bar.ran([C.sub.T])]. Thus it follows that [C.sub.T] has dense range if and only if [C.sub.T][C.sup.*.sub.T] is the operator of multiplication by [f.sub.0] [omicron] T, where [f.sub.0] denotes d([lambda][T.sup.-1])/d[lambda]. Every essentially bounded complex valued measurable function [f.sub.0] induces a bounded operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [L.sup.2]([lambda]), which is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for every f [member of] [L.sup.2]([lambda]). Further [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let us denote d([lambda][T.sup.-1])/d[lambda] by h i.e [f.sub.0] by h and d([lambda][T.sup.-k])/d[lambda] by [h.sub.k], where k is a positive integer greater than or equal to one. Then [C.sup.*.sub.T][C.sub.T] = [M.sub.h] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In general, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the multiplication operator on [L.sup.2]([lambda]) induced by the complex valued measurable function [h.sub.k]. Hyponormal composition operators are studied by Alan Lambert [1]. Paranormal composition operators are studied by T. Veluchamy and S. Panayappan [11].

[section] 3. Definition and properties

Defnition 3.1. An operator T satisfying the condition [[parallel][T.sup.2+k]x[parallel].sup.1/1+k] [[parallel]Tx[parallel].sup.k/1+k] [greater than or equal to] [parallel][T.sup.2]x[parallel], for some integer k [greater than or equal to] 1 and for every x [member of] H is called extended k-paranormal operator or, in short ek-paranormal operators.

If we replace x by [T.sub.x] in the definition of k-paranormal operators, we get ek-paranormal operators. But the converse is not true. This is clear from the following example.

Example 3.2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then T is not k-paranormal for any positive integer k. But T is ek-paranormal.

We characterize ek-paranormal operators as below.

Theorem 3.3. For each positive integer k, an operator T is ek-paranormal if and only if

[T.sup.*2+k][T.sup.2+k] - (1 + k)[[mu].sup.k][T.sup.*2][T.sup.2] + k[[mu].sup.1+k][T.sup.*]T [greater than or equal to] 0 for every [mu] > 0.

Example 3.4. Let H be the direct sum of a denumerable number of copies of two dimensional Hilbert space R x R. Let A and B be two positive operators on R x R. For any fixed positive integer n, define an operator T = [T.sub.A,B,n] on as follows:

T(([x.sub.1],[x.sub.2], ...) = (0, A([x.sub.1]), A([x.sub.2]), ..., A([x.sub.n]),B([x.sub.n+1]), ...).

Its adjoint is [T.sup.*] (([x.sub.1], [x.sub.2], ...)) = (A([x.sub.2]), A([x.sub.3]), ..., A([x.sub.n+1]), B([x.sub.n+2]), ...).

Let n [greater than or equal to] k. Then by Theorem 3.3, T is ek-paranormal if the following conditions are satisfied by A and B.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies the above conditions for every integer k [greater than or equal to] 1. Hence T is ek-paranormal, for every k [greater than or equal to] 1.

Theorem 3.5. If T is ek-paranormal for k = 1, then T is ek-paranormal for every positive integer k.

Proof. Let T be ek-paranormal for k = 1. Then

[parallel][T.sup.4]x[parallel] [[parallel]Tx[parallel].sup.2] [greater than or equal to] [[parallel].sup.3]x[parallel].sup.2]/[parallel][T.sup.2]x[parallel] [[parallel][T.sup.x][parallel].sup.2] [greater than or equal to] [[parallel][T.sup.2]x[parallel].sup.3].

Hence T is ek-paranormal for k = 2. Similarly we can show that if T is ek-paranormal for both k =1 and k = 2, then T is ek-paranormal for k = 3, and so on for every positive integer k.

Theorem 3.6. If T is ek-paranormal and if [alpha] is a scalar, then [alpha]T is also ek-paranormal.

Proof. If [alpha] = 0, the result is trivial. So assume that [alpha] = 0.. Then for any [mu] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence [alpha]T is ek-paranormal.

The following example shows that the ek-paranormal operators are not translation invariant.

Example 3.7. Recall that if H = [C.sup.2], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is ek-paranormal for every positive integer k. But [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not ek-paranormal for any positive integer k.

Theorem 3.8. Let T be ek-paranormal, 0 [not equal to] [lambda] [member of] [[sigma].sub.p](T) and T is of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on ker(T - [lambda]) [direct sum] ker[(T - [lambda]).sup.[perpendicular to]], then

[T.sub.2][T.sub.3](1 + [T.sub.3]/[lambda] + [([T.sub.3]/[lambda]).sup.2] + ... + [([T.sub.3]/[lambda]).sup.k]) = (1 + k)[T.sub.2][T.sub.3].

Proof. Without loss of generality, assume that [lambda] = 1 . Since T is ek-paranormal, taking [mu] = 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A matrix of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and Y = [X.sup.1/2]W[Z.sup.1/2], for some contraction W. Hence we get the required result.

Theorem 3.9. Let T be ek-paranormal, 0 [not equal to] [lambda] [member of] [[sigma].sub.p](T) and T is of the form T = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on ker(T - [lambda]) [direct sum] ker[(T - [lambda]).sup.[perpendicular to]], then [T.sub.3] is also ek-paranormal.

Proof. Using Theorem 3.3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the result [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some contraction W, we get (i) X([mu]) [greater than or equal to] 0, (ii) Z([mu]) [greater than or equal to] 0 and (iii) there exists a contraction W([mu]) such that Y([mu]) = X[([mu]).sup.1/2]W([mu])Z[([mu]).sup.1/2]. Therefore,

[T.sup.*2+k.sub.3][T.sub.3.sup.2+k] - (1 + k)[[mu].sup.k][T.sup.*2.sub.3][T.sub.3.sup.2] + k[[mu].sup.k+1][T.sup.*.sub.3][T.sub.3] [greater than or equal to] (1 + k)f([mu])/X([mu]) [T.sup.*.sub.3][T.sup.*.sub.2][T.sub.2][T.sub.3],

where f([mu]) = (1 + k)(1 - [[mu].sup.k])X([mu]) [greater than or equal to] 0 for all [mu] [greater than or equal to] 0, since f([mu]) has a minimum value at [mu] = 1 . Hence [T.sub.3] is ek-paranormal.

Theorem 3.10. If T is a ek-paranormal operator and [parallel][T.sub.j][parallel] = [[parallel]T[parallel].sup.j] for some j [greater than or equal to] 2, then T is a normaloid.

Proof. For any j [greater than or equal to] 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence by induction, for all positive integers l, [[parallel]T[parallel].sup.lk+2] = [parallel][T.sup.lk+2][parallel]. Therefore, there exists a subsequence {[T.sup.ni]} of {[T.sup.n]} such that lim [[parallel][T.sup.ni][parallel].sup.1/ni] [??] [parallel]T[parallel]. Hence r(T) = [parallel]T[parallel]. Hence T is normaloid.

Theorem 3.11. If T is ek-paranormal, for some positive integer k, then asc(T) is finite.

Proof. By the definition of the operator, ker [T.sup.k+2] is a subset of [kerT.sup.2], which in turn is a subset of [ker.sup.Tk+1]. Hence [kerT.sup.k+1] = [kerT.sup.k+2] and hence the result.

Theorem 3.12. If T is ek-paranormal for some positive integer k and commutes with an isometric operator S, then ST is ek-paranormal.

Proof. Since S is an isometry, [S.sup.*]S = I. Therefore,

[(ST).sup.*2+k][(ST).sup.2+k] - (1 + k)[[mu].sup.k][(ST).sup.*2](ST) + k[[mu].sup.1+k][(ST).sup.*](ST) = [T.sup.*2+k][T.sup.2+k] - (1 + k)[[mu].sup.k][T.sup.*2][T.sup.2] + k[[mu].sup.1+k][T.sup.*]T [greater than or equal to] 0.

Hence is ek-paranormal.

Theorem 3.13. An operator unitarily equivalent to a ek-paranormal for some positive integer k, is also a ek-paranormal operator.

Proof. Let S be unitarily equivalent to a ek-paranormal operator T, for some positive integer k. Then S = [UTU.sup.*] for some unitary operator U. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence S is also ek-paranormal. Hence the result.

[section] 4. Composition operators of ek-paranormal operators

Theorem 4.1. For each positive integer k, [C.sub.T] is ek-paranormal if and only if

[h.sub.2+k] - (1 + k)[[mu].sup.k][h.sub.2] + k[[mu].sup.1+k]h [greater than or equal to] 0 a.e., for every [mu] > 0.

Proof. [C.sub.T] is ek-paranormal for a positive integer k if and only if

[C.sup.*2+k.sub.T][C.sub.T.sup.2+k] - (1 + k)[[mu].sup.k][C.sup.*2.sub.T][C.sub.T.sup.2] + k[[mu].sup.1+k][C.sup.*.sub.T][C.sub.T] [greater than or equal to] 0, for every [mu] > 0,

if and only if for every f [member of] [L.sup.2]([lambda]),

<[C.sup.*2+k.sub.T][C.sub.T.sup.2+k]f, f> - (1 + k)[[mu].sup.k]([C.sup.*2.sub.T][C.sup.2.sub.T]f, f> + k[[mu].sup.1+k] <[C.sup.*T][C.sub.T]f, f> [greater than or equal to] 0,

if and only if <[h.sub.2+k]f, f) - (1 + k)[[mu].sup.k]([h.sub.2]f, f) + k[[mu].sup.k](hf, f) [greater than or equal to] 0, if and only if for every characteristic function [[chi].sub.E] of E in [SIGMA],

[[integral].sub.X]([h.sub.2+k] - (1 + k)[[mu].sup.k][h.sub.2] + k[[mu].sup.1+k]h)[chi]E[bar.[chi]E]d[lambda] [greater than or equal to] 0,

if and only if [h.sub.2+k] - (1 + k)[[mu].sup.k][h.sub.2] + k[[mu].sup.1+k]h [greater than or equal to] 0 a.e. for every [mu] > 0.

Corollary 4.2. [C.sub.T] is ek-paranormal for a positive integer k if and only if [h.sup.k+1.sub.2] [less than or equal to] [h.sub.2+k][h.sup.k] a.e.

[section] 5. Weighted composition operators and Aluthge transformation of k-paranormal operators

A weighted composition operator induced by T is defined as W f = w(f [omicron] T), is a complex valued [SIGMA] measurable function. Let [w.sub.k] denote w(w [omicron] T)(w [omicron] [T.sup.2]) ... (w [omicron] [T.sup.k-1]). Then [W.sup.k] f = [w.sub.k][(f [omicron] T).sup.k] [9]. To examine the weighted composition operators effectively Alan Lambert [1 associated conditional expectation operator E with T as E(x/[T.sup.-1][SIGMA]) = E(x). E(f) is defined for each non-negative measurable function f [member of] [L.sup.p](p [greater than or equal to] 1) and is uniquely determined by the conditions

1. E(f) is [T.sup.-1] [SIGMA] measurable.

2. if B is any [T.sup.-1] [SIGMA] measurable set for which [[integral].sub.B] fd[lambda] converges, we have [[integral].sub.B]fd[lambda] = [[integral].sub.B] E(f)d[lambda].

As an operator on [L.sup.p], [SIGMA] is the projection onto the closure of range of T and E is the identity operator on [L.sup.p] if and only if [T.sup.-1] [SIGMA] = [SIGMA]. Detailed discussion of E is found in [6], [12] and [7].

The following proposition due to Campbell and Jamison is well-known.

Proposition 5.1.[2] For w [greater than or equal to] 0,

1. [W.sup.*]Wf = h[E([w.sup.2])] [omicron] [T.sup.-1] f.

2. [WW.sup.*]f = w(h [omicron] T)E(wf).

Since [W.sup.k]f = [w.sub.k](f [Omicron] [T.sub.k]) and [W.sup.*k]f = [h.sub.k]E([w.sub.k]f)[omicron][T.sup.-k], we have [W.sup.*k][W.sup.k] = [h.sub.k]E([w.sup.2.sub.k])[omicron][T.sup.-k] f, for f [member of] [L.sup.2]([lambda]). Now we are ready to characterize k-paranormal weighted composition operators.

Theorem 5.2. Let W [member of] B([L.sup.2]([lambda])). Then W is ek-paranormal if and only if [h.sub.k+2]E([w.sup.2.sub.k+2]) [omicron] [T.sup.-(k+2)] - (1 + k)[[mu].sup.k][h.sub.2]E([w.sup.2.sub.2]) [omicron] [T.sup.-2] + k[[mu].sup.1+k]hE([w.sup.2]) [omicron] [T.sup.-1] [greater than or equal to] 0 a.e, for every [mu] > 0.

Proof. Since W is ek-paranormal,

[W.sup.*2+k][w.sup.2+k] - (1 + k)[[mu].sup.k][W.sup.*2][W.sup.2] + k[[mu].sup.1+k][W.sup.*]W [greater than or equal to] 0, for every [mu] > 0.

Hence

[[integral].sub.E][h.sub.k+2]E([w.sup.2.sub.k+2]) [omicron] [T.sup.-(k+2)] - (1 + k)[[mu].sup.k][h.sub.2]E([w.sup.2.sub.2]) [omicron] [T.sup.-2] + k[[mu].sup.1+k]hE([w.sup.2]) [omicron] [T.sup.-1] d[lambda] [greater than or equal to] 0

for every E [member of] E and so

[h.sub.k+2]E([w.sup.2.sub.k+2]) [omicron] [T.sup.-(k+2)] - (1 + k)[[mu].sup.k][h.sub.2]E([w.sup.2.sub.2]) [omicron] [T.sup.-2] + k[[mu].sup.1+k]hE([w.sup.2]) [omicron] [T.sup.-1] [greater than or equal to] 0 a.e. for every [mu] > 0.

Corollary 5.3. Let [T.sup.-1][SIGMA] = [SIGMA]. Then W is ek-paranormal if and only if [h.sub.k+2][w.sup.2.sub.k+2] [omicron] [T.sup.-(k+2)] - (1 + k)[[mu].sup.k][h.sub.2][w.sup.2.sub.2] [omicron] [T.sup.-2] + k[[mu].sup.1+k]h[w.sup.2] [omicron] [T.sup.-1] [greater than or equal to] 0 a.e. for every [mu] > 0.

The Aluthge transformation of T is the operator [??] given by [??] = [[absolute value of T].sup.1/2] U [[absolute value of T].sup.1/2]. It was introduced by Aluthge [2]. More generally we may form the family of operators [T.sub.r] : 0 < r [less than or equal to] 1 where [T.sub.r] = [[absolute value of T].sup.r] U [[absolute value of T.sup.1-r] [3]. For a composition operator C, the polar decomposition is given by C = U[absolute value of C] where [absolute value of C] f = [square root of h]f and U f = 1/[square root of h[omicron]T] f [omicron] T. Lambert [5] has given a more general Aluthge transformation for composition operators as [C.sub.r] = [[absolute value of C].sup.r] U [[absolute value of C].sup.1-r] as [C.sub.r] f = [(h/h[omicron]T).sup.r/2] f [omicron] T. i.e [C.sub.r] is weighted composition with weight [pi] = [(h/h[omicron]T).sup.r/2].

Corollary 5.4. Let [C.sub.r] [member of] B([L.sup.2]([lambda])). Then [C.sub.r] is of ek-paranormal if and only if [h.sub.k+2]E([[pi].sup.2.sub.k+2]) [omicron] [T.sup.-(k+2)] - (1 + k)[[mu].sup.k][h.sub.2]E([[pi].sup.2.sub.2]) [omicron] [T.sup.-2] + k[[mu].sup.k+1]E([[pi].sup.2]) [omicron] [T.sup.-1] [greater than or equal to] 0 a.e. for every [mu] > 0.

References

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[2] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integral Equations and Operator theory, 13(1990), 307-315.

[3] A. Aluthge, Some generalized theorems on p-hyponormal operators for 0 < p < 1, Integral Equations and Operator theory, 24(1996), 497-502.

[4] T. Ando, Operators with a norm condition, Acta Sci. Math.(Szeged), 33(1972), 169-178.

[5] Charles Burnap, K. Bong Jung and Alan Lambert, Separating partial normality classes with composition operators, J. Operator theory, 53(2005), 381-397.

[6] J. Campbell and J. Jamison, On some classes of weighted composition operators, Glasgow Math. J, 32(1990), 82-94.

[7] S. R. Foguel, Selected topics in the study of markov operators, Carolina Lectures Series No. 9, Dept. Math., UNC-CH, Chapel Hill, NC. 27514, 1980.

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[9] S. Panayappan, Non-hyponormal weighted composition operators, Indian J. of pure. App . Maths, 27(1996), 979-983.

[10] A. Uchiyama, On the isolated points of the spectrum paranormal operators, Integral Equations and Operator theory, 55(2006), 291-298

[11] T. Veluchamy and S. Panayappan, Paranormal composition operators, Indian J. of pure and applied math, 24(2004), 257-262.

[12] M. Embry Wardrop and A. Lambert, Measurable transformations and centred composition operators, Proc. Royal, Irish acad, 90A(1990), 165-172.

[13] J. Yuan and Z. Gao, Weyl spectrum of class A(n) and n-paranormal operators Integral Equations and Operator theory, 55(2006), 291-298.

S. Panayappan ([dagger]), N. Jayanthi ([double dagger]) and D. Sumathi (#)

Post Graduate and Research Department of Mathematics, Government Arts College (Autonomous), Coimbatore. 18, Tamilnadu, India E-mail: [e-mail]

(1) This work is supported by the UGC, New Delhi (Grant F. No: 34-148/2008(SR)).