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Generalised Weyl and Weyl type theorems for algebraically [k.sup.*]-paranormal operators.

[section]1. Introduction and preliminaries

Let B(H) be the Banach algebra of all bounded linear operators on a non-zero complex Hilbert space H. By an operator T, we mean an element in B(H). If T lies in B(H), then [T.sup.*] denotes the adjoint of T in B(H). The ascent of T denoted by p(T), is the least nonnegative integer n such that ker [T.sup.n] = ker [T.sup.n+1] . The descent of T denoted by q(T) is the least non-negative integer n such that ran([T.sup.n]) = ran([T.sup.n+1]). T is said to be of finite ascent if p(T - [lambda]) < [infinity], for all [lambda] [member of] C. If p(T) and q(T) are both finite then p(T) = q(T) ([11], Proposition 38.3), Moreover, 0 < p([lambda]I - T) = q([lambda]I - T) < [infinity] precisely when A is a pole of the resolvent of T.

An operator T is said to have the single valued extension property (SVEP) at [[lambda].sub.0] [member of] C, if for every open neighborhood U of [[lambda].sub.0], the only analytic function f : U [right arrow] X which satisfies the equation ([lambda]I - T)f([lambda]) = 0 for all [lambda] [member of] U is the function f [equivalent to] 0. An operator T is said to have SVEP, if T has SVEP at every point [lambda] [member of] C. An operator T is called a Fredholm operator if the range of T denoted by ran(T) is closed and both ker T and ker [T.sup.*] are finite dimensional and is denoted by T [member of] [PHI](H). An operator T is called upper semi-Fredholm operator, T [member of] [[PHI].sub.+] (H), if ran(T) is closed and ker T is finite dimensional. An operator T is called lower semi- Fredholm operator, T [member of] [[PHI].sub.-](H), if ker [T.sup.*] is finite dimensional. The index of a semi-Fredholm operator is an integer defined as ind(T) = dim ker T - dim ker [T.sup.*]. An upper semi-Fredholm operator, with index less than or equal to 0 is called upper semi-Weyl operator and is denoted by T [member of] [[PHI].sup.- .sub.+](H).

A lower semi-Fredholm operator with index greater than or equal to 0 is called lower semi-Weyl operator and is denoted by T [member of][PHI][+.bar](H). A Fredholm operator of index 0 is called Weyl operator.

An upper semi-Fredholm operator with finite ascent is called upper semi-Browder operator and is denoted by T [member of] [B.sub.+](H) while a lower semi-Fredholm operator with finite descent is called lower semi-Browder operator and is denoted by T [member of] [B.sub.-] (H). A Fredholm operator with finite ascent and descent is called Browder operator. Clearly, the class of all Browder operators is contained in the class of all Weyl operators. Similarly the class of all upper semi-Browder operators is contained in the class of all upper semi-Weyl operators and the class of all lower semi-Browder operators is contained in the class of all lower semi-Weyl operators. An operator T is Drazin invertible, if it has finite ascent and descent.

For an operator T and a non-negative integer n, define [T.sub.[n]] to be the restriction of T to R([T.sup.n]) viewed as a map from R([T.sup.n]) into R([T.sup.n]). In particular, [T.sub.[0]] = T. If for some integer n, R([T.sup.n]) is closed and [T.sub.[n]] is an upper(resp. a lower) semi-Fredholm operator, then T is called an upper(resp. lower) semi-B-Fredhom operator. Moreover if [T.sub.[n]] is a Fredholm operator, then T is called a B-Fredholm operator. A semi-B-Fredholm operator is an upper or a lower semi-B-Fredholm operator. The index of a semi-B-Fredholm operator T is the index of semi-Fredholm operator [T.sub.[d]], where d is the degree of the stable iteration of T and defined as d = inf{n [member of] N; for all m [member of] N, m [greater than or equal to] n [right arrow] (R([T.sup.n])[intersection] N(T)) [subset] (R([T.sup.m]) [intersection] N(T))}. T is called a B-Weyl operator if it is B-Fredholm of index 0.

The spectrum of T is denoted by [sigma](T), where

[sigma](T) = {[lambda] [member of] C : T - [lambda]I is not invertible}.

The approximate point spectrum of T is denoted by [[sigma].sub.a] (T), where

[[sigma].sub.a](T) = {[lambda] [member of] C : T - [lambda]I is not bounded below}.

The essential spectrum of T is defined as

[[sigma].sub.e] (T) = {[lambda] [member of] C : T - [lambda]I is not Fredholm}.

The essential approximate point spectrum of T is defined as

[[sigma].sub.ea](T) = {[lambda] [member of] C : T - [lambda]I [not member of] [PHI][bar.+](H)}.

The Weyl spectrum of T is defined as

w(T) = {[lambda] [member of] C : T - [lambda]I is not Weyl}.

The Browder spectrum of T is defined as

[[sigma].sub.b](T) = {[lambda] [member of] C : T - [lambda]I is not Browder}.

The set of all isolated eigenvalues of finite multiplicity of T is denoted by [[pi].sub.00](T) and the set of all isolated eigenvalues of finite multiplicity of T in [[sigma].sub.a](T) is denoted by [[pi].sup.a.sub.00](T). [p.sub.00](T) is defined as [p.sub.00](T) = [sigma](T) - [[sigma].sub.b](T). E(T) denotes the isolated eigenvalues of T with no restriction on multiplicity.

The B-Weyl spectrum [[sigma].sub.BW] (T) of T is defined by

[[sigma].sub.BW] (T) = {[lambda] [member of] C : T - [lambda]I is not a B-Weyl operator}.

We say that Weyl's theorem holds for T [8] if T satisfies the equality

[sigma](T) - w(T) = [[pi].sub.00](T)

and a-Weyl's theorem holds for T [17] if T satisfies the equality

[[sigma].sub.a](T) - [[sigma].sub.ea](T)= [[pi].sup.a.sub.00](T).

We say that T satisfies generalized Weyl's theorem [5] if

[sigma](T) - [[sigma].sub.BW](T) = E(T).

We say that T satisfies property (w) if

[[sigma].sub.a](T) - [[sigma].sub.ea](T) = [[pi].sub.00](T)

and T satisfies property (b) if

[[sigma].sub.a](T) - [[sigma].sub.ea](T) = [p.sub.00](T).

By [7], if Generalized Weyl's theorem holds for T, then Weyl's theorem holds for T.

An operator T is called normaloid if r(T) = [parallel]T[parallel] , where r(T) = sup{[absolute value of [lambda]] : [lambda] [member of] [sigma](T)}. An operator T is called hereditarily normaloid, if every part of it is normaloid.

An operator T is called polaroid if iso [sigma](T) [subset or equal to] [pi](T), where [pi](T) is the set of poles of the resolvent of T and iso[sigma](T) is the set of all isolated points of [sigma](T). An operator T is said to be isoloid if every isolated point of [sigma](T) is an eigenvalue of T. An operator T is said to be reguloid if for every isolated point [lambda] of [sigma](T), [lambda]I - T is relatively regular. An operator T is known as relatively regular if and only if ker T and T(X) are complemented. Also Polaroid [??] reguloid [??] isoloid.

K. S. Ryoo and P. Y. Sik defined [k.sup.*]-paranormal operators in [18], k being a positive integer and showed that [sup.*]-paranormal operators form a proper subclass of [k.sup.*]]-paranormal operators for k [greater than or equal to] 3, and [k.sup.*]-paranormal operators are normaloid.

In this paper, we prove that [k.sup.*]-paranormal operators have H property and if 0 [not equal to] A is an isolated point of the spectrum of [k.sup.*]-paranormal operator T for a positive integer k, then the Riesz idempotent operator E of T with respect to [lambda] satisfies [E.sub.[lambda]]H = ker(T - [lambda]) = ker[(T - [lambda]).sup.*]. We also show that if T is [k.sup.*]-paranormal operator, then T is polaroid and Weyl's theorem hold for both T and [T.sup.*]. If in addition [T.sup.*] has SVEP, then a-Weyl's theorem hold for both T and [T.sup.*] and also for f (T) for every f [member of] H([sigma](T)), the space of all analytic functions on an open neighborhood of spectrum of T.

We define algebraically [k.sup.*]-paranormal operators and prove that if T is algebraically [k.sup.*]-paranormal, then Generalised Weyl's theorem hold for T and Weyl's theorem hold for T and f (T), for every f [member of] H([sigma](T)), T is polaroid and hence has SVEP. We prove that if either T or [T.sup.*] is algebraically [k.sup.*]]-paranormal, then spectral mapping theorem holds for essential approximate point spectrum of T. Other Weyl type theorems are also discussed.

[section]2. Definition and properties

In this section, we charecterise [k.sup.*]-paranormal operators and using Matrix representation, we prove that the restriction of [k.sup.*]-paranormal operators to an invariant subspace is also [k.sup.*]-paranormal and ker(T - [lambda]) [subset] ker[(T - [lambda]).sup.*].

Definition 2.1.[18] An operator T is called [k.sup.*]-paranormal for a positive integer k, if for every unit vector x in H, [parallel][T.sup.k]x[parallel] [greater than or equal to][[parallel][T.sup.*]x[parallel].sup.k].

Example 2.1. 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,A,n] on H on as follows:

T(([x.sub.1],[x.sub.2],[x.sub.3],...)) = (0, A([x.sub.1]), A([x.sub.2]),...,A([x.sub.n]),B([x.sub.n+1]),...).

Its adjoint [T.sup.*] is given by

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

Let A and B are positive operators satisfying [A.sup.2] = C and [B.sup.4] = D, where C = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and D = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then T = [T.sub.A,Bn] is of [k.sup.*]-paranormal for k=1.

Theorem 2.1.[18] For k [greater than or equal to] 3, there exists a [k.sup.*]-paranormal operator which is not [sup.*]- paranormal operator.

Theorem 2.2.[18] If T is a [k.sup.*]-paranormal operator, then T is normaloid.

Theorem 2.3. An operator T is [k.sup.*]-paranormal for a positive integer k if and only if for any [mu] > 0,

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

Proof. Let [mu] > 0 and x [member of] H with [parallel]x[parallel] = 1. Using arithmetic and geometric mean inequality, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conversely assume that [T.sup.*k][T.sup.k] - k[[mu].sup.k-1]T[T.sup.*] + (k - 1)[[mu].sup.k] [greater than or equal to] 0.

If [parallel][T.sup.*]x[parallel] = 0, then the [k.sup.*]-paranormality condition is trivially satisfied.

If x [member of] H with [parallel][T.sup.*]x[parallel] [not equal to] 0 and [parallel]x[parallel] = 1, taking [mu] = [[parallel][T.sup.*]x[parallel].sup.2] , we get

[parallel][T.sup.k]x[parallel] [greater than or equal to] [[parallel][T.sup.*]x[parallel].sup.k].

Hence T is [k.sup.*]-paranormal.

Theorem 2.4. If T [member of] B(H) is a [k.sup.*]-paranormal operator for a positive integer k, T does not have a dense range and T has the following representation: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on H = [bar.ran(T)][direct sum] [C] ker([T.sup.*]), then [T.sub.1] is also a [k.sup.*]-paranormal operator on [bar.ran(T)] and [T.sub.3] = 0. Furthermore, [sigma](T) = [sigma]([T.sub.1])[union] {0}, where [sigma](T) denotes the spectrum of T.

Proof. Let P be the orthogonal projection onto [bar.ran(T)]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since T is of [k.sup.*]-paranormal operator, by Theorem 2.3,

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

Hence

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

Hence

[T.sup.*k.sub.1] - [T.sup.k.sub.1]-k[[mu].sup.k][T.sub.1][T.sup.*.sub.1] + (k - 1)[[micro].sup.k-1] [[absolute value of [T.sup.*.sub.2]].sup.2] [greater than or equal to]0.

Hence [T.sub.1] is also [k.sup.*]-paranormal operator on [bar.ran(T)].

Also for any x = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

0. Hence [T.sub.3] = 0. By ([10], Corollary 7), [sigma]([T.sub.1])[union][sigma]([T.sub.3]) = [sigma](T)[union][tau], where t is the union of certain of the holes in [sigma](T) which happen to be a subset of [sigma]([T.sub.1])[intersection] [sigma]([T.sub.3]), and [sigma]([T.sub.1])[intersection][sigma]([T.sub.3]) has no interior points. Therefore [sigma](T) = [sigma]([T.sub.1])[union] [sigma]([T.sub.3]) = [sigma]([T.sub.1])[union] {0}.

Theorem 2.5. If T is [k.sup.*]-paranormal operator for a positive integer k and M is an ivariant subspace of T, then the restriction [T.sub.|M] is [k.sup.*]-paranormal.

Proof. Let P be the orthogonal projection onto M. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since T is of [k.sup.*]-paranormal operator, by Theorem 2.3,

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

Hence

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

Hence

[T.sup.*k.sub.1] - [T.sup.k.sub.1]-k[[mu].sup.k][T.sub.1][T.sup.*.sub.1] + (k - 1)[[micro].sup.k-1] [[absolute value of [T.sup.*.sub.2]].sup.2] [greater than or equal to]0.

Hence [T.sub.1], i.e., [T.sub.|M] is also [k.sup.*]-paranormal operator on M.

Theorem 2.6. If T is [k.sup.*]-paranormal operator for a positive integer k, 0 [not equal to] [lambda] [member of] [[sigma].sub.p](T) and

T is of the form T = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(i) [T.sub.2] = 0,

(ii) [T.sub.3] is [k.sup.*]-paranormal.

Proof. Let T = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Without loss of generality assume that [lambda] = 1. Then by Theorem 2.3 for [mu] = 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A matrix of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if X [greater than or equal to] 0, Z [greater than or equal to] 0 and Y = [X.sup.1/2] W[Z.sup.1/2], for some contraction W. Therefore [T.sub.2] [T.sup.*.sub.2] = 0 and [T.sub.3] is [k.sup.*]-paranormal.

Corollary 2.1. If T is [k.sup.*]-paranormal operator for a positive integer k and (T - [lambda])x = 0 for [lambda] [not equal to] 0 and x [member of] H, then [(T - [lambda]).sup.*] x = 0.

Corollary 2.2. If T is [k.sup.*]-paranormal operator for a positive integer k, 0 [not equal to] [lambda] [member of] [[sigma].sub.p](T), then T is of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and ker([T.sub.3] - [lambda]) = {0}.

[section]3. Spectral properties

If [lambda] [member of] iso [sigma](T), the spectral projection (or Riesz idempotent) [E.sub.[lambda]] of T with respect to [lambda] is defined by [E.sub.[lambda]] = [1/2[pi]i][[integral].sub.[partial derivative]D[(z - T).sup.-1] dz, where D is a closed disk with centre at [lambda] and radius small enough such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this section, we show that [k.sup.*]-paranormal operators have (H) property and if [lambda] [member of] [sigma](T) is an isolated point, then [E.sub.[lambda]] with respect to [lambda] is self-adjoint and satisfies [E.sub.[lambda]]H = ker(T - [lambda]) = ker[(T - [lambda]).sup.*]. Weyl's theorem hold for both T and [T.sup.*] and if [T.sup.*] has SVEP, then a-Weyl's theorem hold for both T and [T.sup.*].

Theorem 3.1. If T is [k.sup.*]-paranormal operator for a positive integer k and for [lambda] [member of] C, [sigma](T) = [lambda] then T = [lambda].

Proof. If [lambda] = 0, then since [k.sup.*]-paranormal operators are normaloid ([18], Theorem 9), T = 0. Assume that [lambda] [not equal to] 0. Then T is an invertible normaloid operator with [sigma](T) = [lambda]. [T.sub.1] = [1/[lambda]]T is an invertible normaloid operator with [sigma]([T.sub.1]) = {1}. Hence [T.sub.1] is similar to an invertible isometry B (on an equivalent normed linear space) with [sigma](B) = 1 ([12], Theorem 2). [T.sub.1] and B being similar, 1 is an eigenvalue of [T.sub.1] = [1/[lambda]]T ([12], Theorem 5). Therefore by theorem 1.5.14 of [14], [T.sub.1] = I. Hence T = [lambda].

Theorem 3.2. If T is [k.sup.*]]-paranormal operator for some positive integer k, then T is polaroid.

Proof. If [lambda] [member of] iso [sigma](T) using the spectral projection of T with respect to [lambda], we can write T = [T.sub.1] [direct sum] [T.sub.2] where [sigma]([T.sub.1]) = {[lambda]} and [sigma]([T.sub.2]) = [sigma](T) - {[lambda]}. Since [T.sub.1] is [k.sup.*]-paranormal operator and [sigma]([T.sub.1]) = {[lambda]}, by Theorem 3.1, [T.sub.1] = [lambda]. Since [lambda] [not member of] [sigma]([T.sub.2]), [T.sub.2] - [lambda]I is invertible. Hence both [T.sub.1] - [lambda]I and [T.sub.2] - [lambda]I and hence T - [lambda]I have finite ascent and descent. Hence [lambda] is a pole of the resolvent of T. Hence T is polaroid.

Corollary 3.1. If T is [k.sup.*]-paranormal operator for some positive integer k, then T is reguloid.

Corollary 3.2. If T is [k.sup.*]-paranormal operator for some positive integer k, then T is isoloid.

Theorem 3.3. If T is [k.sup.*]-paranormal operator for a positive integer k and [lambda] [sigma](T) is an isolated point, then the Riesz idempotent operator E[lambda] with respect to [lambda] satisfies [E.sub.[lambda]]H = ker(T - [lambda]). Hence [lambda] is an eigenvalue of T.

Proof. Since ker(T - [lambda]) [subset or equal to] [E.sub.[lambda]]H, it is enough to prove that [E.sub.[lambda]]H [subset or equal to] ker(T - [lambda]). Now [sigma]([T.sub.|E[lambda]H]) = {[lambda]} and [T.sub.|E[lambda]H] is [k.sup.*]-paranormal. Therefore by Theorem 3.1, [T.sub.|E[lambda]H] = [lambda]. Hence E[lambda]H = ker(T - [lambda]).

Theorem 3.4. Let T be a [k.sup.*]-paranormal operator for a positive integer k and [lambda] [not equal to] 0 be an isolated point in [sigma](T) . Then the Riesz idempotent operator [E.sub.[lambda]] with respect to A is self-adjoint and satisfies [E.sub.[lambda]]H = ker(T - [lambda]) = ker[(T - [lambda]).sup.*].

Proof. Without loss of generality assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [lambda]) [direct sum] ran[(T - [lambda]).sup.*]. By Theorem 2.6, [T.sub.2] = 0 and [T.sub.3] is [k.sup.*]-paranormal. Since 1 [member of] iso [sigma](T), either 1 [member of] iso [sigma]([T.sub.3]) or 1 [not member of] [sigma]([T.sub.3]). If 1 [member of] iso [sigma]([T.sub.3]), since [T.sub.3] is isoloid, 1 [member of] [[sigma].sub.p] ([T.sub.3]) which contradicts ker([T.sub.3]- [lambda]) = {0} (by Corollary 2.2). Therefore 1 [not member of] [sigma]([T.sub.3]) and hence [T.sub.3] - 1 is invertible. Therefore T - 1=0 [direct sum] ([T.sub.3] - 1) is invertible on H and ker(T - 1) = ker[(T - 1).sup.*]. Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore [E.sub.[lambda]] is the orthogonal projection onto ker(T - [A.sub.)] and hence [E.sub.[lambda]] is self-adjoint.

Let T [member of] L(X) be a bounded operator. T is said to have property (H) if [H.sub.0]([lambda]I - T) = ker([lambda]I - T), where [H.sub.0](T) = {x [member of] X : [lim.sub.n[right arrow][infinity]] [[parallel][T.sup.n] x[parallel].sup.1/n] = 0}. By [13], [E.sub.[lambda]]H = [H.sub.0]([lambda]I - T). Hence by Theorem 3.3, [k.sup.*]-paranormal operators have (H) property. Hence by Theorems 2.5, 2.6 and 2.8 of [3], we get the following results:

Theorem 3.5. If T is [k.sup.*]-paranormal operator for some positive integer k, then T has SVEP, p([lambda]I - T) [less than or equal to ] 1 for all [lambda] [member of] C and [T.sup.*] is reguloid.

Theorem 3.6. If T is [k.sup.*]-paranormal operator for some positive integer k, then Weyl s theorem holds for T and [T.sup.*]. If in addition, [T.sup.*] has SVEP, then a-Weyl s theorem holds for both T and [T.sup.*].

Theorem 3.7. If T is [k.sup.*]-paranormal operator for some positive integer k and [T.sup.*] has SVEP, then a-Weyl's theorem holds for f (T) for every f [member of] H([sigma](T)).

[section]4. Algebraically [k.sup.*]-paranormal operators

In this section, we prove spectral mapping theorem and essential approximate point spectral theorem for algebraically [k.sup.*]-paranormal operators and also show that they are polaroids.

Definition 4.1. An operator T is defined to be of algebraically [k.sup.*]-paranormal for a positive integer k, if there exists a non-constant complex polynomial p(t) such that p(T) is of class [k.sup.*]-paranormal.

If T is algebraically [k.sup.*]-paranormal operator for some positive integer k, then there exists a non-contant polynomial p(t) such that p(T) is [k.sup.*]-paranormal. By the Theorem 3.5, p(T) is of finite ascent. Hence p(T) has SVEP and hence T has SVEP ([14], Theorem 3.3.6).

Theorem 4.1. If T is algebraically [k.sup.*]-paranormal operator for some positive integer k and [sigma](T) = [[mu].sub.0], then T - [[mu].sub.0] is nilpotent.

Proof. Since T is algebraically [k.sup.*]-paranormal there is a non-constant polynomial p(t) such that p(T) is [k.sup.*]-paranormal for some positive integer k, then applying Theorem 3.1,

[sigma](p(T)) = p([sigma](T)) = {p([[mu].sub.o])} implies p(T) = p([[mu].sub.o]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are invertible, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence T - [[mu].sub.0]

Theorem 4.2. If T is algebraically [k.sup.*]-paranormal operator for some positive integer k, then w(f(T)) = f (w(T)) for every f [member of] H ol([sigma](T)).

Proof. Suppose that T is algebraically [k.sup.*]-paranormal for some positive integer k, then T has SVEP. Hence by ([11], Proposition 38.5), ind(T - [lambda]) [less than or equal to ] 0 for all complex numbers [lambda]. Now to prove the result it is sufficient to show that f(w(T)) [subset or equal to] w(f(T)). Let [lambda] [member of] f(w(T)). Suppose if [lambda] [not member of] w(f (T)), then f (T) - [lambda]I is Weyl and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since each of ind(T - [[lambda].sub.i]) [less than or equal to ] 0, we get that ind(T - [[lambda].sub.i])= 0, for all i = 1, 2, ..., n. Therefore T - [[lambda].sub.i] is weyl for each i = 1, 2,..., n. Hence [[lambda].sub.i] [not member of] w(T) and hence [lambda] [not member of] f (w(T)), which is a contradiction. Hence the theorem.

Theorem 4.3. If T or [T.sup.*] is algebraically [k.sup.*]-paranormal operator for some positive integer k, then [[sigma].sub.ea](f (T)) = f ([[sigma].sub.ea](T)).

Proof. For T [member of] B(H), by [16] the inclusion [[sigma].sub.ea] (f (T)) [subset or equal to] f ([[sigma].sub.ea](T)) holds for every f [member of] H ([sigma](T)) with no restrictions on T. Therefore, it is enough to prove that f ([[sigma].sub.ea](T))[subset or equal to][[sigma].sub.ea] (f(T)).

Suppose if [lambda] [not member of] [[sigma].sub.ea] (f (T)) then f (T) - [lambda] [member of] [[PHI].sup.-.sub.+](H), that is f (T) - [lambda] is upper semi-Fredholm operator with index less than or equal to zero. Also f (T)- [lambda] = c(T-[[alpha].sub.1])(T-[[alpha].sub.2])...(T-[[alpha].sub.n])g(T), where g(T) is invertible and c, [[alpha].sub.1], [[alpha].sub.2],...,[[alpha].sub.n] [member of] C.

If T is algebraically [k.sup.*]-paranormal for some positive integer k, then there exists a nonconstant polynomial p(t) such that p(T) is [k.sup.*]-paranormal. Then p(T) has SVEP and hence T has SVEP. Therefore ind(T - [[alpha].sub.i]) [less than or equal to ] 0 and hence T - [[alpha].sub.i] [member of] [[PHI].sup.-.sub.+](H) for each i =1, 2,...,n. Therefore [lambda] = f ([[alpha].sub.i]) [not member of] f ([[sigma].sub.ea](T)). Hence [[sigma].sub.ea] (f (T)) = f ([[sigma].sub.ea](T)).

If [T.sup.*] is algebraically [k.sup.*]-paranormal for some positive integer k, then there exists a nonconstant polynomial p(t) such that p([T.sup.*]) is [k.sup.*]-paranormal. Then p([T.sup.*]) has SVEP and hence [T.sup.*] has SVEP. Therefore ind(T - [[sigma].sub.i]) [greater than or equal to] 0 for each i = 1, 2,...,n. Therefore 0 [less than or equal to ] [[summation].sup.n.sub.i=1]ind(T - [[alpha].sub.i]) = ind(f (T) - [lambda]) [less than or equal to ] 0. Therefore ind(T - [[alpha].sub.i]) = 0 for each i = 1, 2,...,n. Therefore T - [[alpha].sub.i] is Weyl for each i = 1, 2,...,n. (T - [[alpha].sub.i]) [member of] [[PHI].sup.-.sub.+](H) and hence [[alpha].sub.i] [not member of] [[alpha].sub.ea](T). Therefore [lambda] = f ([[alpha].sub.i]) [not member of] f ([[alpha].sub.ea](T)) . Hence [[alpha].sub.ea] (f (T)) = f ([[alpha].sub.ea](T)).

Theorem 4.4. If T is algebraically k -paranormal operator for some positive integer k, then T is polaroid.

Proof. If [lambda] [member of] iso [sigma](T) using the spectral projection of T with respect to [lambda], we can write T = [T.sub.1] [direct sum][T.sub.2] where [sigma]([T.sub.1]) = {[lambda]} and a([T.sub.2]) = [sigma](T) - {[sigma]}. Since [T.sub.1] is algebraically [k.sup.*]-paranormal operator and [sigma]([T.sub.1]) = {[lambda]}, by Theorem 4.1, [T.sub.1] - [lambda]I is nilpotent. Since [lambda] [not member of] [sigma]([T.sub.2]), [T.sub.2] - [lambda]I is invertible. Hence both [T.sub.1] - [lambda]I and [T.sub.2] - [lambda]I and hence T - [lambda]I have finite ascent and descent. Hence [lambda] is a pole of the resolvent of T. Hence T is polaroid.

Corollary 4.1. If T is algebraically [k.sup.*]-paranormal operator for some positive integer k, then T is reguloid.

Corollary 4.2. If T is algebraically [k.sub.*]-paranormal operator for some positive integer k, then T is isoloid.

[section]5. Generalised Weyl's theorem and other Weyl type theorems

In this section, we prove Generalised Weyl's theorem for algebraically [k.sup.*]]-paranormal operators and discuss other Weyl type theorems.

Theorem 5.1. If T is algebraically [k.sup.*]-paranormal operator for some positive integer k, then generalized Weyl s theorem holds for T.

Proof. Assume that [lambda] [member of] [sigma](T) - [[sigma].sub.BW](T), then T - [lambda] is B-Weyl and not invertible. Claim: [lambda] [member of] [partial derivative][sigma](T). Assume the contrary that [lambda] is an interior point of [sigma](T). Then there exists a neighborhood U of [lambda] such that dimN(T - [mu]) > 0 for all [mu] in U. Hence by ([9], Theorem 10), T does not have SVEP which is a contradiction. Hence [lambda] [member of] [partial derivative][sigma](T) - [[sigma].sub.BW] (T). Therefore by punctured neighborhood theorem, [lambda][member of] E(T).

Conversely suppose that [lambda] [member of] E(T). Then [lambda] is isolated in [sigma](T). Using the Riesz idempotent E[lambda] with respect to [lambda], we can represent T as the direct sum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then by Theorem 4.1, [T.sub.1] - [lambda] is nilpotent. Since [lambda] [not member of] [sigma]([T.sub.2]), [T.sub.2] - [lambda] is invertible. Hence both [T.sub.1] - [lambda] and [T.sub.2] - [lambda] have both finite ascent and descent. Hence T - [lambda] has both finite ascent and descent. Hence T - [lambda] is Drazin invertible. Therefore, by ([6], Lemma 4.1), T - [lambda] is B-Fredholm of index 0. Hence [lambda] [member of] [sigma](T) - [[sigma].sub.BW] (T). Therefore [sigma](T) - [[sigma].sub.BW] (T) = E(T).

Corollary 5.1. If T is algebraically [k.sup.*]-paranormal operator for some positive integer k, then Weyl s theorem holds for T.

By ([4], Theorem 2.16) we get the following result: Corollary 5.2. If T is algebraically [k.sup.*]-paranormal for some positive integer k and [T.sup.*] has SVEP then a-Weyl s theorem and property(w) hold for T.

Theorem 5.2. If T is algebraically [k.sup.*]-paranormal operator for some positive integer k, then Weyl's theorem holds for f (T), for every f [member of] Hol([sigma](T)).

Proof. For every f [member of] H([sigma](T)),

[sigma](f(T)) - [[pi].sub.00] (f(T)) = f([sigma](T) - [n.sub.00](T)) by ([15], Lemma) = f(w(T)) by Theorem 5.2. = w(f(T)) by Theorem 4.3.

Hence Weyl's theorem holds for f (T), for every f [member of] H([sigma](T)).

If [T.sup.*] has SVEP, then by ([1], Lemma 2.15), [[sigma].sub.ea] (T) = w(T) and [sigma](T) = [[sigma].sub.a](T). Hence we get the following results:

Corollary 5.3. If T is algebraically [k.sup.*]-paranormal for some positive integer k and if in addition [T.sup.*] has SVEP, then a-Weyl's theorem holds for f (T) for every f [member of] H([sigma](T)).

Corollary 5.4. If [T.sup.*] is algebraically [k.sup.*]-paranormal for some positive integer k, then w (f(T))= f(w(T)).

By ([1], Theorem 2.17), we get the following results:

Corollary 5.5. If T is algebraically [k.sup.*]-paranormal for some positive integer k and [T.sup.*] has SVEP then property (b) hold for T.

Corollary 5.6. If T is algebraically [k.sup.*] -paranormal for some positive integer k, Weyl's theorem, a-Weyl's theorem, property (w) and property (b) hold for [T.sup.*].

References

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

Post Graduate and Research Department of Mathematics, Government Arts College, Coimbatore 18, India

E-mail: Jayanthipadmanaban@yahoo.in
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Date:Mar 1, 2012
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