On Some n-Involution and k-Potent Operators on Hilbert Spaces

In this paper, we survey various results concerning n -involution operators and k -potent operators in Hilbert spaces. We gain insight by studying the operator equation n T I = , with , 1 k T I k n ≠ ≤ − where , n k ∈N . We study the structure of such operators and attempt to gain information about the structure of closely related operators, associated operators and the attendant spectral geometry. The paper concludes with some applications in integral equations.


Introduction
for all x H ∈ ) (see [5] for more exposition). Two operators S and T are said to be nearly equivalent if there exists an invertible operator V such that It is well known that ( ) r T is equal to the actual radius of the spectrum, that is, Behaviour of the powers n T of a linear operator T on a Hilbert space H has been studied by some researchers, leading to important applications. It is well-known that linear operators and their powers may behave quite differently. T is a contraction then T is similar to a contraction C , which implies that n C I = which in turn implies that C is unitary (by use of the Nagy-Foias-Langer Decomposition for contractions ([5], § 5.1)). Operators of the form 2 ki n T e I π = provide the simplest example of ninvolution operators in Hilbert spaces. It has been shown in ( [3], Theorem 1) that all n -involutions are of this form. Clearly an n -involution need not be unitary although it is norm-preserving and invertible. Note that if T is an ninvolution then 1 T − and * T are n -involutions (see [1], [6], [10]). We may have operators T such that n T I = , with Clearly if T is n -involution then [6]) and all the previous results hold. We note that an n -involution need not be unitary.

Main Results
Thus T is an invertible isometry and is therefore a unitary operator.  The following two results follow from definitions. Proposition 2.9 Let S be a self-adjoint involution which is metrically equivalent to T . If T is self-adjoint then it is an involution. Proposition 2.10 Let The converse is trivial. Theorem 2.12 If T is an n-involution, then r(T) = 1.
The result below gives a condition when 1 n T = implies Note that if T is normal then T is a normal n-involution then 1 T = . This follows from the fact that 1 n n T T = = .
Theorem 2.13 If T is a normal n-involution then 1 T = .
Proof. This follows from the fact that 1 Theorem 2.13 also holds when normal is replaced by normaloid operator.
Theorem 2.14 If T is an n-involution, then T is n-normal, for some n ∈ ℕ .
Proof. This follows easily from Example 2. The operator with matrix 0 1 shows that T is 4-normal.
Definition 2.15 Let C be a simple smooth closed oriented curve and let ( ) t α be a one-to-one mapping of C onto itself.
The function ( ) t α is called a shift function or simply a shift Evidently α(t) is a generalized Carleman shift of order n, preserving the orientation of C. The function 1 Carleman shift of order 2 on C, changing the orientation of C.

Spectral Properties of n-Involutions
In this section we study some spectral properties of ninvolutions.
Recall that a complex number λ is said to be unimodular

Proposition 3.2 If T is an n-involution then every component of the spectrum of T intersects the unit circle.
Proof. Since By the Jordan canonical decomposition, the matrix T is similar to a block-diagonal matrix Theorem 3.4 can be generalized as follows.

Theorem 3.5 Let T be an n-involution. A scalar λ ∈ σ(T) if
and only if |λ| = 1. An operator T is called an n-symmetry if it is a unitary ninvolution, that is, An operator T is an n-reflection if Example 3.
The operators 1 1 acting on 2 C and 3 C , respectively are 2 - reflections since a simple calculation shows that 2 2 , The following claims follow easily from definitions.
Theorem 3.7 An operator ( ) T B H ∈ is an n-involution if and only if T is similar to a diagonal matrix , Note also that if T is an n-involution then Clearly the following operator class inclusion holds where the bar denotes complex conjugation and t R ∈ is an involution. Clearly The reflection operator Clearly reflections are self-adjoint involutions.

n-Involutions, Associated Idempotents, k-Potents and Geometry
Involutions have a wide range of applications in geometry. Interestingly, there is a close relationship between ninvolutions and idempotent operators. Proposition 4.1 (a). If T is an involution then 1 ( ) 2 P I T = + is an idempotent operator.

(b). If P is an idempotent operator then T=2P-I is an involution.
Proof.
(a). 2 the class of involutions and that of idempotent operators. We shall call the involution T and the idempotent P associated. Proposition 4.2 If P is a rank one idempotent operator then the operators 2 , , 0 I P P I P P I P I

an involution if and only if it is the difference of a pair of complementary idempotent operators.
Proof. Suppose that 2 2 , , Then P − Q = T and it is easy to see that this decomposition is unique: Example 5. The projection as its associated involution. Remark. An idempotent operator need not be self-adjoint. We give a condition under which it is self-adjoint in terms of its associated involution.

Proposition 4.5 An involution is self-adjoint if and only if its associated idempotent is self-adjoint.
Proof. Let T be a self-adjoint involution. Then * = (b). T is normal and 2 T I = .
(c). There exists an orthogonal projection such that 1 ( ) Proof. We first note that assertion (a) says that T is a selfadjoint unitary or a symmetry.
is called a dilatation in the ratio λ. In the special case λ = 1, the dilatation becomes the identity operator. If λ = 0, T is a projection and if λ = −1, then T is an involution (indeed, a unitary involution, if P is an orthogonal projection).
An invection T is a linear operator satisfying 4 T I = . That is, T is a 4-involution.

Proposition 4.8 Every involution is an invection.
Remark. We note that the converse of Proposition 4.8 is not true in general. There exist non-involutory 4-involutions.

A I = , then A is similar to an operator with matrix
We note that in Theorem 4.9 similarity cannot be replaced with unitary equivalence.
Let   I T T T  T T  T  T  T   I T  T  T   I T T T   P = + + + + + + + + For more recent exposition on tripotent operators (see [8], [10]  Note that the converse to Proposition 4.17 is not generally true. For instance an involution T cannot be a 3-involution, unless T = I.

(c). If T is a k-potent for 2 k ≥ , then Ran(T) is closed and
The following result is trivial.

Proposition 4.18 If T is an n-involution then n T is an involution.
Theorem 4.19 If T is an involution with associated idempotent P, then Ran(T) = Ker(I − P) and Ran(P) = Ker(I − T). Proof.
x P I x The second part follows by substituting 1 ( ).
). Proof. From Theorem 4.14, every 4-involution is decomposable as a product of two involutions. Invoking Theorem 4.21, the claim follows. Theorem 4.23 Every idempotent operator T is k-potent, for every integer 2 k ≥ . Theorem 4.23 asserts that every projection operator is kpotent, for every integer k ≥ 2. But not every k-potent, for every integer k > 2 is necessarily a projection.
The following results show that if two idempotent operators have equal range then they are similar.  Proof. The fact that T is a unitary involution implies that

Proposition 4.28 An operator T is an n-involution if and only if * T is an n-involution. Theorem 4.29 Let T be a 2-isometry. If T is an involution, then T is unitary.
Proof. Using Proposition 4.28 and a simple computation gives Thus T is an invertible isometry, which must be unitary. This proves the claim.

Theorem 4.30 Let T be a self-adjoint operator. Then T is an involution if and only if T is unitary.
Proof. Suppose is an involution which is not an isometry (and hence not a symmetry). Remark. It has been shown by Singh et al [8] that the product of two tripotent operators is a tripotent operator if and only if they commute. The sum of involutions need not be an involution, even if they commute. However, it is clear that every n-involution is a sum of two complementary idempotent operators and that every n-involution is an n + 1potent operator. We aver that the claim by Singh etal [8] is equivalent to saying that the product of two commuting involutions is a tripotent operator.
Operators A and B are said to quasicommute if AB BA T = + , where T is a compact operator (see also [4]    Remark. Wilf-equivalence is an equivalence relation stronger than Q-equivalence.
Question. Does Q-equivalence preserve self-adjointness, invertibility, norm, numerical range, etc. of operators? How is it related to other operator equivalence relations?
It is clear that Q-equivalence preserves invertibility but it does not preserve norm, spectrum, and self-adjointness of operators and numerical range of operators. To see this, let 0 1 . This example also reveals that Q-equivalent operators need not have equal spectra, even if Q is unitary. If Q is unitary, then Remark. We note that Q-equivalence of operators is weaker than similarity. To see this, suppose A = QB. Proof. We prove the converse since the other direction has been proved in the remark above. Suppose, without loss of generality that A = Q(BQ) 1 Q − . Then a simple computation shows that A = QB. This proves the claim. The converse of Theorem 4.41 is not generally true. The unilateral shift and identity operators on 2 ( ) ℓ ℕ are metrically equivalent but not U-equivalent for any unitary operator U. We note that the converse of the above statement holds in finite dimensional Hilbert spaces. Proof. Follows from the proof of Proposition 4.41.

Discussion
The notion of an operator T satisfying n T I = is applicable in solving singular integral equations with a Carleman shift that involve an involutive operator Q such that 2 Q I = , or more generally, n Q I = .  1 2 , , , n A A A ⋯ are bounded linear operators in a Banach space under consideration (see [4]). Projection operators (associated with involutions) are useful in vast areas of physics-in quantum theory, many-body physics, applications in group theory, projective geometry, statistical mechanics of irreversibility, to mention but a few. Quadratic forms with idempotent and tripotent operator matrices are extensively used in the theory of statistics, especially in the area of multivariate normal distributions (see [9]).

Conclusion
In this paper, the structure of some n -involution and kpotent operators and their relationships has been shown. It has been shown that any normaloid n -involution is unitary. It has been shown that unitary equivalence, similarity and quasisimilarity preserve the n -involutory property of operators and that metric equivalence preserves this property for self-adjoint operators. Several conditions under which an n -involution has norm one has been proved. The notion of Q -equivalence is introduced and it is shown that if two ninvolutions are U -equivalent for some unitary operator then they have the same norm.