Table of Contents
Declaration and Approval ii
Dedication iii
Acknowledgments v
Chapter 1 Introduction
1.1 Notations, Terminologies and Definitions 1
1.2 Normal and Non-normal operators 9
1.3 Historical development of Hilbert Spaces 11
1.4 Historical development of Fuglede-putnam’s theorem 12
1.5 Series of inclusion of classes of operators 14
Chapter 2 Normality of non normal operators
2.1 Some general properties of normal and non normal operators 16
2.2 Normality of hyponormal,p-hyponormal,log-hyponormal,subnormal and semi-normal op- erators 23
2.2.1 Normality of n-power normal operators and n-power quasinormal operators 26
2.2.2 Normality of other non-normal operators 29
Chapter 3 ON THE FUGLEDE-PUTNAM THEOREM
Chapter 4 Applications and conclusions
4.1 Applications of Fuglede-Putnam theorem 42
4.2 Conclusion 45
4.3 Summary 47
4.4 Open Problem 48
Bibliography 49
Chapter 1
Introduction
In this chapter, We give terminologies, notations and definitions that will be used through- out the project and also some brief historical backgrounds of some normal and non- normal operators.
1.1 Notations, Terminologies and Definitions
Notations
H: Hilbert space over C
B(H): Banach algebra of bounded operators
T ∗: the adjoint of an operator T
ǁTxǁ: the operator norm of T
ǁxǁ: the norm of a vector x
σ (T ): the spectrum of an operator T
π0(T ) : point spectrum of an operator T
Ran(T ) : the range of an operator T
Ker(T ):the kernel of an operator T
M ⊕ M⊥: the direct sum of the Subspaces M and M⊥ of H.
Terminologies and Notations
Throughout this paper, H or K will denote a complex Hilbert space and B(H) will denote the Banach algebra of bounded linear operators on H
We denote the kernel and range of an operator T by Ker(T ) and Ran (T ) respectively. M and M⊥ stands for the closure and orthogonal complement of of a closed subspace M of H.
We denote by σ (T ), π0(T ), ǁT ǁ and W (T ) the spectrum, point spectrum, norm and nu- merical range of T ∈ B(H) respectively.
We write w(T ) for the Weyle spectrum of T and w(T ) for the closure of w(T ).
We denote the essential numerical range of T by We(T ) and the set of all isolated points of the spectral of T that are eigenvalues of finite multiplicity by σ00(T ).
Let M be a closed subspace of H and T ∈ B(H) be an operator. We denote the restriction of T to M by T |M.
Definition 1.1.1 An inner product space is a vector space E together with a map< . >:
E × E → F such that
(i) < λx + µy, z >= λ < x, z > +µ < y, z >
(ii) < x, y >= < y, x >
(iii) < x, x >≥ 0, < x, x >= 0 if and only if x = 0∀x, y, z ∈ E and λ ∈ F.
Definition 1.1.2 let X be a vector space and ǁ.ǁ : X → R··· ∗ be a real valued function. Then the function ∗ is called a norm if it satisfies the following.
(i) ǁxǁ ≥ 0 and ǁxǁ = 0 if and only if x = 0
(ii)ǁλxǁ = |λ |ǁX ǁ∀x ∈ X, λ ∈ R
(iii) ǁx + yǁ ≤ ǁxǁǁyǁ∀x, y ∈ X .
The pair (X, x ) is called a normed space.
Proposition 1.1.3 (Cauchy-Schwarz’s inequality)
For any two elements x, y in an inner product space X,
|< x, y >| ≤ ǁxǁǁyǁ.
Definition 1.1.4 An operator T ∈ B(H) is said to be normal if it commutes with its ad- joint(i.e, T ∗T = TT ∗, equivalently, T ∗T TT ∗ = 0)
Preposition 1.1.5 Let T be an operator on a Hilbert Space H. The following assertions
are equivalent
(i) T is normal
(ii) ǁT ∗xǁ = ǁTxǁ for any x ∈ H
(iii) T is normal for any integer ,n 1
(iv) T ∗nx 2 = Tnx 2.
Remark 1.1.6 By relaxing some conditions of normality of operators, we obtain non normal operators. The results below we check some of these nonnormal operators. definition 1.1.7 An operator T ∈ B(H) is said to be quasinormal if T (T ∗T ) = (T ∗T )T i.e(T ∗T TT ∗)T = 0.
Proposition 1.1.8 A unilateral and bilateral shi operators are quasinormal operators.
Definition 1.1.9 An operator T ∈ B(H) is said to be subnormal if it has a normal extension.
Proposition 1.1.10 Every Subnormal operator T ∈ B(H) on a finite dimensional Hilbert Space is normal.
Definition 1.1.11 An operator T ∈ B(H) is said to be hyponormal if T ∗T ≥ TT ∗.
Proposition 1.1.12 An operator T ∈ B(H) is hyponormal if and only if ǁT ∗xǁ ≤ ǁTxǁ.
=⇒ If T is hyponormal, then TT ∗ ≤ T ∗T if and only if
< TT ∗x, x >≤< T ∗Tx, x > i.e < T ∗x, T ∗x >≤< Tx, Tx >
that is ǁT ∗xǁ2 ≤ ǁTxǁ2 which implies
ǁT ∗xǁ ≤ ǁTxǁ.
Definition 1.1.13 An operator T ∈ B(H) is said to be hyponormal if its adjoint T ∗ is hyponormal.
That is, TT ∗ ≥ T ∗T .
Proposition 1.1.14. An Operator T ∈ B(H) is normal if and only if it is both hyponormal and cohyponormal.
Definition 1.1.15. An operator T ∈ B(H) is semi-normal if it is either hyponormal or cohyponormal.
Definition 1.1.16 An operator T ∈ B(H) is said to be paranormal if T x ≥ ǁTxǁ for
Definition 1.1.17 An operator T is called normaloid if
ǁT ǁ = sup| < Tx, x > |
ǁ ǁ
.
Definition 1.1.18 An operator T is called convexoid if the closure of the numerical range equals the convex hull of the spectrum of T .
That is
W (T ) = {< Tx, x >: ǁxǁ = 1} = σ (T )
.
Definition 1.1.19. An operator T ∈ B(H) is n-normal if TnT ∗ = T ∗Tn.
Remark 1.1.20. The class of all n-normal operators is denoted by [nN].
Definition 1.1.21. An operator T ∈ B(H) is said to be binormal if T ∗T commutes with TT ∗.
Definition 1.1.22. An operator T is said to be w-hyponormal if |∆(T )| ≥ |T | ≥ |∆∗(T )|.
Definition 1.1.23. An operator T ∈ B(H) is said to be spectraloid if W (T ) = r(T ). Definition 1.1.24. An operator T ∈ B(H) is said to be a scalar if it is a scalar multiple of the identity operator (i.e T = αI, α ∈ C).
Definition 1.1.25 An operator T is said to be an isloid if any isolated point of δ (T ) is an eigenvalue of T .
Definition 1.1.26 An operator T ∈ B(H) is said to be p-hyponormal if (T ∗T )p ≥ (TT ∗)p. Definition 1.1.27.An operator T ∈ L(H) is posinormal if there exists a positive operator P ∈ B(H) such that TT ∗ = T ∗PT .
Definition 1.1.28 An operator T L(H) is coposinormal if T ∗ is posinormal.
Remark 1.1.29. Every hyponormal operator is posinormal but the converse is not generally true.
Proposition1.1.30 Let T be a posinormal operator. Then T is hyponormal if and only if KerT = KerT ∗.
Definition 1.1.31. An operator T ∈ B(H) is said to be log-hyponormal if T is invertible and log T ∗T ≥ logTT ∗.
Definition 1.2.32 An operator T ∈ L(H) is said to be p-posinormal if (TT ∗)P ≤ α(T ∗T )p for some α > 1 and P > 0.
Definition 1.1.33. An operator T ∈ B(H) is said to be semi-hyponormal if (T2∗T )2 ≥
Definition1.1.34. An operator T ∈ L(H) is said to be polaroid if every isolated point of the spectrum of T is a pole of the resolvent of T .
Definition 1.1.35. An operator is said to be an adjoint of an operator T if there exists a unique operator T ∗ ∈ B(K, H) such that < Tx, y >=< x, T ∗y > ∀x ∈ H, y ∈ (K). In this case,T ∗ is called the adjoint of T .
Theorem 1.1.36. For S, T ∈ B(H, K), the following holds
(i) αS + T )∗ = α¯ S∗ + T ∗
(ii)(s∗)∗ = S
(iii(ST )∗ = T ∗S∗
(iv) I∗ = I,where I is the identity operator in H
(v) ǁT ∗xǁ = ǁT ǁ2 hence ǁT ∗ǁ = ǁT ǁ.
Definition 1.1.37. An operator T ∈ B(H) is said to be Hermitian (or self-adjoint ) if T = T ∗.
Remark 1.1.38 Every Hermitian Operator is normal.
Definition 1.1.39. An operator P ∈ B(H) is said to be idempotent if P2 = P. Definition 1.1.40. An operator P ∈ B(H) is said to be a projection if P is idempotent and Ker(P) = Ran(P)⊥.
Theorem 1.1.41. Let P ∈ B(H) be an idempotent operator. Then the following properties are equivalent
(i) P is a projection
(ii) P is the orthogonal projection of H onto Ran P
(iii) ǁPǁ = 1
(iv) P is Hermitian
(v) P is normal
(vi) P is positive
Definition 1.1.42 An operator U ∈ B(H) is said to be an isometry if ǁUxǁ = ǁxǁ∀x ∈ H.
Theorem 1.1.43 For an operator U ∈ B(H), the following are equivalent.
(i) U is an isometry
(ii) U ∗U = I,the identity in H
(iii) < Ux,Uy >=< x, y > x, y H
Proof
(i) =⇒ (ii) 2
∀x ∈ H, < (U ∗U − IH)x, x >= ǁUxǁ
Thus U ∗U − IH = 0
which implies U ∗U = IH
(ii) =⇒ (iii)
< Ux,Uy >=< U ∗Ux, y >=< x, y >
(iii) = (i)
2— ǁxǁ = 0
ǁUxǁ =< Ux,Ux >=< x, x >= ǁxǁ
Definition 1.1.44 An operator U ∈ B(H, K) is said to be a partial isometry if it satisfies the following conditions.
(i) U = UU ∗U
(ii) P = U ∗U is a projection
(iii) U |Ker⊥U is an isometry.
Definition 1.1.45. An operator T ∈ B(H) is said to be co-isometry if TT ∗ = I (that is ,T is a co-Isometry if T ∗ is an Isometry).
Definition 1.1.46. An operator U ∈ B(H) is called unitary if T ∗T = TT ∗ = I.
Remark 1.1.47. An operator U ∈ B(H, K) is unitary if and only if U is both an isometry and co-isometry.
Remark 1.1.48. A unitary operator is normal.
definition 1.1.49. An operator T ∈ B(H) is said to be an essential isometry if T ∗T − I is compact.
Definition 1.1.50. An operator T ∈ B(H) is said to be an essential co-isometry if TT ∗ − I is compact.
definition 1.1.51. An operator t ∈ B(H) is said to be positive if ∀x, y ∈ H, < Tx, y >≥ 0 and T is Hermitian.
Remark 1.1.52. Every projection operator is positive
Definition 1.1.53 An operator T ∈ B(H, K) is invertible if there exists an operator S ∈
B(K, H) such that ST = I and TS = I.
Remark 1.1.54 An invertible operator is denoted by T −1.
Proposition 1.1.55. For S and T invertible operators, then the following equality holds true.
(TS)−1 = S−1T −1
Proof
(TS)(S−1T −1) = T (SS−1)T −1 = TIT −1 = TT −1 = I
and
(S−1T −1)(TS) = S−1(T −1T )S = S−1IS = S−1S = I
Remark 1.1.56 A unitary operator is invertible.
Definition 1.1.57 An operator T ∈ B(H) is said to compact if for every bounded sequence,{xn} in H, the sequence {Txn} has a subsequence which converges in H.
Definition 1.1.58 An operator T ∈ B(H, K) is said to be a Hilbert-Schmidt operator if it satisfies the following conditions.
(i) ∑n ǁTenǁ < ∞ for some orthonormal basis {en} of H
(ii) ∑m ǁT ∗ fmǁ2 < ∞ for some orthonormal basis { fm} in K
(iii) ∑n ǁTenǁ < ∞ for all orthonormal basis {en} of H.
Definition 1.1.59. An operator T ∈ B(H) is called Fredholm if there exists an operator S such that the operators STI and TSI are compact
Remark 1.1.60 Let T be a Fredholm operator. The index of T denoted by IndT is defined by IndT = dimKerT dimKerT ∗.
Remark 1.1.61 If T and S are Fredholm operators, then TS is a Fredholm operator and IndTS = IndT + IndS.
Definition 1.1.62 An operator S on H is called (Unilateral)shi operator if Sen = en + 1, n = 1, 2 · · · for some orthonormal basis {en} of H.
definition 1.1.63 An operator T ∈ B(H) is called anti-adjoint (equivalently skew Hermi- tian ) if T ∗ = −T .
Definition 1.1.64 An operator T ∈ B(H) is called a semi-shi operator if
(i) T is an Isometry
(ii) ∩∞ RanT = 0.
definition 1.1.65. An operator T is said to be Backward shi if it satisfies the following conditions
(i) dim (kerT ) = I.
(ii) The Induced operator Tˆ : x/kerT → X defined by Tˆ(x + KerT ) = Tx is an isometry.
(iii) ∪n=1∞KerTn is dense in X
Definition 1.1.66 An operator T ∈ B(H) is called a le shi operator if Tx = y where
x = (x1, x2 · · · ) and y = (x2, x3 · · · ) ∈ l2.
Definition 1.1.67. An operator T ∈ B(H) is called a right shi operator if T (x1, x2, · · · ) = (0, x1, x2, · · · ) ∈ l2.
Definition 1.1.68 An operator V for a function f ∈ l2[0, 1] and a value t ∈ [0, 1] defined
Remark 1.1.69 V is bounded. We note that V is a Hilbert-Schmidt operator and hence in particular compact.
Definition 1.1.70. The rank of an operator T L(H, K) is the dimension of range of T .
Definition 1.1.71 An operator T L(H) is a finite rank operator if Ran(T) is finite dimension.
Remark 1.1.72 A finite rank operator need not be bounded.
Proposition 1.1.73 If an Operator T is bounded, linear and has a finite rank, then T is compact.
Remark 1.1.74. Every T ∈ B(H) finite rank operator is compact.
Definition 1.1.75. Suppose 1 ≤ P ≤ ∞, the Hardy space Hp is defined by Hp = { f ∈ L (T ) f (n) = 0, n < 0 .
Remark 1.1.76 H2 endowed with the l2-scalar product is a Hilbert space with an orthonormal basis.
Definition 1.1.77 An operator Tα is said to be Toeplitz if Tα : H2 H2, f P(α f ) where p is the projection of l2 ontoH2 and α l∞(T ).
Remark 1.1.78. A Toeplitz operator is self adjoint
Definition 1.1.79 An operator T ∈ B(H) is said to be involutive if if T 2 = I.
Definition 1.1.80 An operator T ∈ B(H) is said to be a contraction if ǁT ǁ ≤ 1. Definition 1.1.81. An operator T ∈ B(H) is called diagonalisable if there exists an or- thonormal basis en for H consisting of eigenvectors of T .
Remark 1.1.82 Any unitary operator on a finite dimensional complex Hilbert Space is diagonalisable.
Definition 1.1.83 An operator T ∈ B(H) is called a numeroid if W (T ) is a spectral set for T .
Definition 1.1.84 An operator T ∈ B(H) is said to be co-subnormal if its adjoint is sub- normal.
Definition 1.1.85 An operator T ∈ B(H) is called co-paranormal if its adjoint is paranormal.
Definition 1.1.86 Operator radius of an operator T ∈ B(H) is defined by wp(T ) = inf{U :
U > 0,U −1T ∈ Cp}, 0 < p < ∞.
Definition 1.1.87 An operator T ∈ B(H) is said to be of class Cp(p ≥ 0) if there exists a unitary operator U on B ∈ B(K) such that Tnx = p pUnx for n = 1, 2 · · · , x ∈ H. Definition 1.1.88 An operator T ∈ B(H) is called p − oid if W (TK) = (W (T ))k, k = 1, 2, · · · .
Remark 1.1.891 − oid and 2 − oid operators are normaloids and spectraloid operators respectively.
Definition 1.1.90 An operator T ∈ B(H) is called a p-convexoid if W (T ) = conv.σ (T ). Definition 1.1.91 An operator T ∈ B(H) is said to be reduction-p if the restriction of T to every invariant subspace of T has property p.
Definition 1.1.92 The Aluthge transformation of an operator T ∈ B(H) (denoted by T˜) is defined as ˜ 1 1
T = |T | 2 U |T | 2 for a unitary operator U.
Definition 1.1.93 An operator T ∈ B(H) is called n-power quasinormal if TnT ∗T =
T ∗TTn = T ∗Tn+1.
Remark 1.1.94 The class of n-power quasinormal operators is denoted by [nQN].
Remark 1.1.95 When n = 1, an n-power quasinormal operator is quasinormal TBm(T ) = T ∑m (−1)kT ∗m−kTm−k = 0
k=0 k
where Bm(T ) is obtained from the binomial expansion of Bm(T ) = (T ∗T − I)m.
Remark 1.1.97 When m = 1, T is called partial isometry
Definition 1.1.98 An operator T ∈ B(H) is called dominant if Ran(A − λ I) ⊆ Ran(A −λ I)∗∀λ C.
Definition 1.1.99 An operator T ∈ B(H) is called Browder if T is Fredholm and T − λ I is invertible for sufficiently small λ /= 0 ∈ C.
Definition 1.1.100 The essential spectrum of T (denoted by σe(T )) is defined by σe(T ) = {λ ∈ C : T − λ I is not Fredholm}.
1.1.101 The Browder spectrum (denoted by σb(T )) of T is defined by
σb(T ) = {λ ∈ C : T − λ I is not Browder}.
Definition 1.1.102 An operator T ∈ B(H) is said to be resuloid if T − λ I is regular for each λ ∈ Isoσ (T ).
Definition 1.1.103 An operator T ∈ B(H) is said to be closoid if Ran(T − λ I) is closed for each λ ∈ Isoσ (T ).
Definition 1.1.104 An operator T ∈ B(H) is called (α, β ) − normal(0 ≤ α ≤ 1 ≤ β ) if α2T ∗T ≤ TT ∗ ≤ β 2T ∗T .
Definition 1.1.105 An operator T ∈ B(H) is said to be m-hyponormal if there exists a positive number m such that
m2(T − λ I)∗(T − λ I) − (T − λ I)(T − λ I)∗ ≤ 0∀λ ∈ C.
Definition 1.1.106 An operator T ∈ B(H) is said to be quasi-invertible if T has zero ker- nel and dense range.
Definition 1.1.107 An operator T ∈ B(H) is called transloid if aT + bI is normaloid a, b C.
Remark 1.1.108 Every Transloid is convexoid
Definition 1.1.109 An operator T ∈ B(H) is said to be a class yα operator(for α ≥ 0) if there exists a positive number kα such that
|TT ∗ − T ∗T |α ≤ K2α (T − α)∗(T − α)∀λ ∈ C.
Remark 1.1.110 A class y operator id m-hyponormal.
Definition 1.1.111 An operator N is called Julia operator (denoted by J(N)) if(1 − NN∗)1 N
J(N) = −N∗ (1 − N∗ )1
Definition 1.1.112 An operator T ∈ B(H) is called p-quasihyponormal if T ∗ ((T ∗T )p − (TT ∗)p) T ≥ 0 for p ≥ 0.
Remark 1.1.113 If p = 1, Then T is quasihyponormal
If p = 1 ,then T is semi-quasihyponormal.
2 ˜ p
Definition 1.1.114 An operator A is said to be p − w−hyponormal operator if |A| ≥
|A|p ≥ |A˜∗|p.
Definition 1.1.115 Let A, B ∈ B(H) be operators. We define the generalized deriva- tion(denoted by δA,B(X )) induced by A and B as
δA,B(X ) = AX − XB∀X ∈ B(H).
Remark 1.1.116 The class of Hilbert-Schmidt operators is denoted by C2(H).
Remark 1.1.117 C2(H) is a Hilbert space .The Hilbert-Schmidt norm of X ∈ C2(H) is given by ǁxǁ2 =< X, X > 2
An operator A is said to be (p, k)−quasiposinormal if A∗ C (A∗A) − (AA∗) A ≥ 0 for some positive integer 0 < p ≤ 1, some C > 0 and a positive integer k.
Definition 1.1.119 Two operators A, B ∈ B(H, K) are said to be similar if there exists an invertible operator N ∈ B(H, K) such that NA = BN.
Definition 1.1.120 Two operators A, B ∈ B(H, K) are said to be unitarily equivalent if there exists a positive unitary operator U ∈ B(H, K) such that UA = BU .
Definition 1.1.121 A subspace M ⊆ H is said to be invariant under T ∈ B(H) if TM ⊆ M. Definition 1.1.122 A subspace M ⊆ H is said to be a reducing subspace of T ∈ B(H) if it is invariant under both T and T ∗.
1.2 Normal and Non-normal operators
Normal operators
The study of normal operators has been very successful in the sense that a lot of inter- esting results have been obtained concerning these operators.
One of the main results of these operators is the classical Fuglede-Putnam theorem that we will discuss in detail in this research paper and the spectral theorem that only holds for normal operators.
Many authors have defined new classes of operators by making them satisfy certain known properties of normal operators in the hope that some of the results which holds for normal operators will also hold for these new classes of operators. This has led to a new area of research on non-normal operators which are as a result of relaxing normality of normal operators.
It is well known that given two normal operators A, B ∈ B(H), A + B, and AB are not normal in general. For example, consider the operators
A simple computation shows that A and B are normal but AB is not.
The question on characterizing those pairs of normal operators for which their products are normal was studied for finite dimensional spaces by Gantmaher and Krein[20] in 1930 and Weigmann[63] for compact operators. However, it is important to point that normality of AB does not always imply normality of BA.
In 1953, Kaplansky[27] showed that under the compactness assumptions, the normality of AB and BA are equivalent. Later on, KiNaneh considered this question in [31] and showed that it is sufficient to assume that A and B∗ be hyponormal and that AB be com- pact in order to conclude that BA is normal.
Gheondea proved the Gantmaher-Krein-Weigmann theorem in [21].
On the normality of any pair of normal operators, it is well known that if each of two normal operators commute with the adjoint of each other, then their sum is normal and so is their product. That is, if A and B are normal operators such that A commute with B∗, then A + B is normal and so is AB.
Yadav and Ramanujan[66] proved that if the real part of each of two normal operators commutes with the imaginary part of the other, then their sum is normal. Mortad also showed in [40] that for two bounded operators A, B ∈ B(H), A + B is normal if AB and AB∗ are normal such that A is positive.
In 1970, Embry[15] introduced the concept of similarities of normal operators by stating that:
if S and T are two commuting normal operators and AS = TA, where 0 ∈/ W (A) for A ∈ B(H), then S = T .
Mortad [41] generalized Embry’s theorem by imposing a self-adjointeness condition on A and dropping the commutativity of S and T and came up with the following result. Let A be bounded self-adjoint operator such that 0 ∈/ W (A). If S and T are bounded nor- mal operators such that AS = TA, then S = T .
Apart from the operations of normal operators, other authors have given more properties of normal operator. For instance,
Putnam[48]has given sufficient condition that a square root of a norma operator is normal. Stampfli[58] also showed a result that an nth root of of an invertible normal operator is similar to a normal operator. Radjavi and Rosenthal[51] gave a clear representation of all square roots of normal operators.
We also note that if T is normal, then any polynomial of T is normal but the converse is not true in general. KiNaneh[32] has shown that if Tn is normal foe some n > 1, then T is quasi-similar to a direct sum of a normal operator and a compact operator. The author also showed that if P(T ) is normal for some nonzero polynomial P and T is essentially normal, then T can be wriNen as a sum of a normal operator and a compact operator.
Thus, the class of normal operators has given birth to various areas of research and many authors today are still trying to relax their normalities to obtain new classes of operators that satisfies some results are only satisfied by normal operators.
Non-normal operators
One of the main results of normal operators is the class of non-normal operators which is achieved by relaxing some normality conditions of normal operators. This class has led to a wide area of research notably the Fuglede-Putnam theorem where many authors have come up with various classes of non-normal which under some conditions, satisfy the Fuglede-Putnam theorem.
As we have seen from section 1.1, there are many classes of non-normal operators and many more are still introduced even today and each of these operators has their own properties which makes them unique from the others. We shall see some of these prop- erties in section 2.
One of the main classes of these operators is the class of subnormal operators. This class was introduced by Hamos in [23] who later defined the concept of hyponormal operators in 1950 by bringing the definition T ∗T ≤ TT ∗. By considering the case where TT ∗ ≤ T ∗T , the class of cohyponormal was thus introduced. This enabled Brown[?] to introduce and study the class of quasinormal operators. As we shall show later on, it was proved that every quasinormal operator is subnormal.
The class of posinormal operators (or positive normal operators) was introduced by Rhaly in [52] and further studied in [24] where the authors studied more properties of this class and showed that Weyl’s theorem holds for some totally posinormal operators.
Jibril[26] introduced the class of n-power normal operators and proved that an operator T ∈ L(H) is n-power normal if and only if ǁTnxǁ = ǁ(Tn)∗xǁ for all x ∈ H and also gave some properties of n-power operators. Alzuraiqi and Patel studied further properties in of this class in [5]. As seen in subsection 1.1, this class is denoted by [nN] for all positive intergers n.
Ahmed [1] continued the work of Jibril by introducing the class of n-power quasinor- mal operators and showed some relations between n-normal and quasi-normal operators. These relations will be investigated in section 2. This class is denoted by [nQN].
1.3 Historical development of Hilbert Spaces
Functional analysis is a very important branch of mathematics that has found numerous applications both in mathematics world and other fields such as engineering and computer science among many others .
One of the cornerstones of Functional analysis is the notion of a Hilbert Space.
Hilbert Space emerged from the German mathematician David Hilbert’s(1862-1943) ef- forts to generalize the concept of Euclidean Space to an infinite dimensional space.
He formulated the theory of square summable spacel2.
However, it would be interesting to note that although Hilbert is considered to be the father of Hilbert Spaces, it was not until years later that von Neumann (1903-1957) gave the definition of a Hilbert space in 1927.
In his work, Neumann formulated an axiomatic theory of Hilbert Space and developed the modern theory of operators in Hilbert spaces .
Although these two great mathematicians contributed highly to the growth and devel- opment of Hilbert spaces, we note that other mathematicians contributed a lot to this development.
The notion of a ’Space’ was introduced by Riemann in his work in 1856 and was also the one who conceived the idea of a closed subspace of a Hilbert space(manifold).
Between 1844 and 1862, Herman Grassman(1809-1877) introduced the concept of a finite dimensional vector space.
Karl Heinstrass (1815-1897)considered the distance between two functions in the context of the calculus of variations but it was not until 1897 that Jacques Hadamand(1865-1963) gave a boost to Hilbert space theory by connecting the set theoretic ideas of Cantor with the notion of a space of functions.
However it was not until 1906 that Hadamand’s PhD student, Maurice frechet, (1878- 1973) astounded the mathematics world by introducing the concept of a metric space.
In 1916, the notion of a topological space was introduced by Felix Hausdorff (1868-1942) which was a crucial boost to the Hilbert theory. Topological spaces have become widely applicable in functional analysis and their contributions cannot be underrated.
Many other modern mathematicians, the likes of Schmidt, have greatly contributed to what we now love and know as the Hilbert Space Theory. The theory has not only enriched the world of mathematics but has proven extremely useful scientific theories. Hilbert spaces play a central role in analysis, mostly functional, geometry group theory and and number theory among others.
1.4 Historical development of Fuglede-putnam’s theorem
The original paper of Fuglede first appeared in 1950[16] where the author proved the Fu- glede’s theorem. This was in answering a problem posed by John Von Neumann [44] in 1942.
In his theorem, Fuglede was able to prove the following result;
Let A and B be bounded operators on a complex Hilbert space with B being normal. If AB = BA,then AB∗ = B∗A.
However, in 1958, Putnam generalized[49]Fuglede’s theorem by proving the following; If A, B, X are linear operators on a complex Hilbert Space and suppose X and B are nor- mal, B is bounded and BA = AX ,then B∗A = AX ∗.
Berberian[10] proved that the Fuglede theorem was actually equivalent to that of Putnam by a nice operator matrix derivation trick. Thus Fuglede -Putnam theorem was born and it states as follows;
Let A and B be normal operators and X be an operator such such such that AX = XB, then A∗X = XB∗. Berberian was able to relax the hypothesis on A and B by requiring X to be a Hilbert -Schmidt operator(i.e X ∈ C2(H)).
A er the work of these great mathematicians, several authors have relaxed the normality of A and B in the Fuglede-Putnam’s theorem over the years in various ways.
In 1958, Rosenblum[56] gave a simple and clear proof of Fuglede Putnams’ theorem by using lioville’s theorem.
Later on, M. Radjabalipour [50] (1987)showed that Fuglede-Putnam’s theorem holds for hyponormal operators.
In 1994, Cha[13] showed that the hyponormality hypothesis can be replaced by the quasi- hyponormality of A and B∗ under some conditions in the Fuglede-Putnam’s theorem.
B.P. Duggal [14] showed that if A, B∗ are p-hyponormal operators , then A and B satisfy Fuglede-Putnam’s theorem.
In 1997, Lee[33] proved that if if A is p-quasihyponormal operator and B∗ is an invertible p-quasihyponormal operator such that AX=XB for X ∈ C2(H) and |A|1−p . |B−1|1−p ≤ 1, then Fuglede-Putnam theorem holds(that is A∗X = XB∗).
In 1996, Patel[46] proved that for a Hilbert Schmidt operator X and A and B∗ being p- hyponormal operators such that AX = BX , then A∗X = XB∗.
Uchiyama and Tanahashi[62](2002) showed that the Fuglede-Putnam theorem holds for p-hyponormal and log-hyponormal operators.
In 2005, Mecheri[38] showed that Lee’s results remain the same without the condition
¨|A|1−p¨ . ¨|B−1|1−p¨ ≤ 1
The author further showed that Lee’s results remain true for (p, k)−quasihyponormal operators without the additional condition
¨|A|1−p¨ . ¨|B−1|1−p¨ ≤ 1
Kim [28] showed that the result A∗X = XB∗ remains remains valid for an injective (p, k)−quasihyponorm and log-hyponormal operator.
In 2009, Bakiri [9] showed that that if A is an injective (p,k)-quasihyponormal in H and B is a dominant operator in H such that AX = XB for some X ∈ B(H), then A∗X = XB∗.
The author also showed that the above result remains valid for injective (p, k)−quasihyponormal and log hyponormal operators.
Mecheri and Uchiyama [39] showed that normality in the Fuglede-Putnam theorem can be replaced by A and B∗ class operators.
Rashid and Noorani[54] showed that the result by Mecheri and Uchiyama for A and B∗ quasi-class A operators with the additional condition
ǁ|A∗|ǁ . |B|−1 ≤ 1
satisfies the Fuglede-putnam theorem.
As recent as 2012, Bashir et al. [6]proved that the Fuglede-Putnam theorem hold for w- hyponormal operators.
Clearly, Fuglede-Putnam theorem has fascinated many mathematicians in the mathe- matics world and many mathematicians are working day and night to try and relax the normality of A and B in the theorem.
1.5 Series of inclusion of classes of operators
In this section, we set to investigate some classes of operators and show some inclusion relationship of these operators.
In 1962, Stampfli[59] introduced hyponormal operators and was able to show that any normal operator is hyponormal.
In 1978, Campbell and Gupta[12] introduced k-quasinormal operators for some k ∈ C. The authors were able to show that by leNing k = 1, then a hyponormal operator would become a k-quasihyponormal.
It was in 1990 that Aluthge [3] astounded the mathematics world by introducing p- hyponormal operators and illustrated that if we let p = 1 in the definition of a p-hyponormal operator, then we get a hyponormal operator.
Another class of operators is the p-quasihyponormal and by leNing p = 1,we get quasi- hyponormal operators.
Tanahashi[60],in 1999 introduced another class of operators called log-hyponormal op- erators which contains all invertible hyponormal operators. He was able to demonstrate that invertible p-hyponormal operators are log-hyponormal operators.
Then, in 2000, Aluthge et al.[4] was able to generalise both log-hyponormal and p-hyponormal operators to w-hyponormal operators which contains all p-hyponormal operators.
In 2003, Hyoun[28] introduced (p, k)−quasihyponormal operators and showed that if we let p = 1and k = 1, in the definition of (p, k)−quasihyponormal operator, then we get k−quasihyponormal and p−quasihyponormal operators respectively.
We note that a q-quasihyponormal operator is a (p, k)−quasihyponormal since 0 < q < p and thus (p,k)-quasihyponormal operators contain all p-hyponormal operators.
In 2007, Jibril[26] introduced he class of 2-power normal operators and later generalized the class of 2-power normal operators in the class of n-power normal operators[25].
Later on, in 2011, Ahmed[1] generalised the work of Jibrii on n-power normal operators into the class of n-power quasinormal operators and through this he was able to show that every n-power normal operator is n-power quasinormal.
In 2012, Panayan[45] introduced an extension of all normal operators which he later call the n-power class operator.
We therefore have;
Projection⊆Self-Adjoint⊆NormalsubseteqHyponormal.
Normal⊆ asinormal⊆Subnormal⊆Hyponormal⊆m-hyponormal.
Unitary⊆Isometry⊆Partial Isometry⊆Contraction.
Unitary⊆Isometry⊆2-normal⊆Binormal.
Normal⊆ asinormal⊆Subnormal⊆Hyponormal⊆P-hyponormal⊆Log-hyponormal.
Normal⊆Hyponormal⊆P-hyponormal⊆w-hyponormal.
Normal⊆ asihyponormal⊆Subnormal⊆Hyponormal⊆m-hyponormal.
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