Abstract
In this project, we
investigate lattices of subspaces (like invariant, reducing and hyper- invariant subspaces among others) of some
operators in Hilbert spaces and also give a detailed
information on equivalence of operators and show how lattices of equivalent operators
relate to each other.
Table of Contents
Abstract ii
Declaration and Approval iv
Dedication vii
Acknowledgments ix
Chapter 1: Introduction
1.1.1 Notations, Terminologies and Definitions 1
1.2 Some Bounded Operators in Hilbert Spaces 4
Chapter 2: LITERATURE REVIEW
2.1 Development of laNice theory 8
2.2 Invariant subspace problem 9
Chapter 3: ON ELEMENTARY LATTICE THEORY
3.1 LATTICES 11
3.2 Types of LaNices 13
3.3 Invariant subspaces on Hilbert spaces 15
3.4 Reductive operators 18
3.5 Hyper-invariant and Hyper-reducing Subspaces 22
Chapter 4: EQUIVALENCE OF OPERATORS AND SUBSPACE LATTICES
4.1 Unitary and similarity of operators 24
4.2 Almost similarity of operators 26
4.3 asisimilarity of operators 27
4.4 Metric equivalence of operators 28
4.5 Unitarily quasi-equivalence of opertors 31
Chapter 5: CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion 33
5.1.1 Recommendations 33
Bibliography 35
Chapter One
1.1 Introduction
This chapter outlines the notations, terminologies and definitions that will be used through- out this work and also give some basic theory on some operators in Hilbert space.
1.1.1 Notations, Terminologies and Definitions
Notations
H: Hilbert Space over the complex numbers C
B(H): Banach algebra of bounded linear operators onH T : A bounded linear operator
T ∗ : The adjoint of T
ǁT ǁ : The operator norm of T
ǁxǁ: The norm of a vector x
< x, y >: The inner product of x and y on a Hilbert Space H Ran(T ): The range of an operator T
Ker(T ): The kernel of an operator T
M ⊕ N: The direct sum of the SubspaceM and N
{T }′: The commutant of T
PM: the orthogonal projection of H onto M.
Terminologies and Definitions
In this project, H and Kwill denote complex Hilbert spaces which may be finite or infinite dimensional and B(H) will denote the Banach algebra of bounded linear operators.
B(H, K) will denote the set of bounded linear operators from H to K and equipped with the norm.
By an operator, we mean a bounded (i.e continuous) linear transformation with domain H and range a subset of H.
we denote the identity operator by I.
We denote by Ma linear manifold in H as a subset of H which is closed under vector addition and under multiplication by complex numbers.
The spectrum of an operator T ∈ B(H)(denoted by σ (T )) is defined as σ (T ) = {λ ∈ C : T − λ I is not invertible }.
If T ∈ B(H), then T ∗ denotes the adjoint of T while Ker(T ), Ran(T ), M¯ and M⊥ stands for the kernel of T , range of T , closure of M and orthogonal complement of M.
Let A, B ∈ B(H) be operators. The commutant of A and B is given by [A, B] = AB − BA.
Definition 1.1.1 A subspace M ⊆ H is said to be invariant under an operator T ∈ B(H) if TM ⊆ M.
Remark 1.1.2 For an invariant subspace M ⊆ H under T , we say that M is T −invariant.
Definition 1.1.3 A subspace M ⊆ H is said to be a reducing subspace of T ∈ B(H) if it is invariant under both T and T ∗ (equivalently if both M and M⊥ are invariant under T ).
Definition 1.1.4 The commutant of T ∈ B(H) is the set of all operators that commute with T .
Remark 1.1.5 We denote the commutant of T ∈ B(H) by {T }′ and define it as {T }′ = (S ∈ B(H) : ST = TS).
Definition 1.1.6 The double commutant of T ∈ B(H),denoted by {T }′′ is defined by {T }′′ = {A ∈ B(H) : AS = SA, S ∈ {T }′}.
Definition 1.1.7 A subspace M ⊆ H is said to be a hyperinvariant subspace for T ∈ B(H) if SM ⊆ M for each S ∈ {T }′.
Definition 1.1.8 A subspace M ⊆ H is said to be a hyper-reducing subspace for T ∈ B(H) if M reduces every operator in the commutant of T .
Remark 1.1.9 We denote the collection of all hyper-reducing subspaces for T ∈ B(H) by HyperRed(T ).
Definition 1.1.10 Let A be a nonempty set. A binary operation ∗ is called associative if a, b A, (a b) c = a (b c) and commutative if a b = b a.
Definition 1.1.11 Let X be a non-empty set. We define an equivalence relation over X as a relation that satisfies the conditions below( we denote a relation by ∼)
(i)x ∼ x (Reflexive)
(ii) if x ∼ y , then y ∼ x (Symmetric)
(iii) if x ∼ y and y ∼ z, then x ∼ z(Transitivity) ∀x, y, z ∈ X .
Definition 1.1.12 Let X be a non-empty set. Let ≤ be a binary relation on X satisfying ∀x, y, z ∈ X
(i) x ≤ x Reflexive
(ii) if x ≤ y and y ≤ x, then x = y Antisymmetric
(iii) if x ≤ y and y ≤ z, then x ≤ z Transitive
Remark 1.1.13 The Set (X, ≤) which satisfies the conditions of definition 1.1.12 is called a partially ordered set or a poset.
Definition 1.1.14 A hasse diagram is a graphical representation of a poset.
Definition 1.1.15 Let X be a partially ordered set with elements a, b ∈ X . If we have that for every pair of elements a, b ∈ X , either a ≤ b or b ≤ a, then X is said to be totally ordered and is called a chain.
Remark 1.1.16 A chain is a totally ordered poset.
The figure below is a hasse diagram representation of a chain.
Figure 1. A chain
Definition 1.1.17 Let X be a set and A ⊆ X be a nonempty subset of X. An element x ∈ X is called an upper bound for A if a x, a A.
Remark 1.1.18 x is called the least upper bound(denoted by lub(A) or sup(A)) for A if x is the smallest upper bound for A.
Definition 1.1.19 Let X be a set and A ⊆ X be a nonempty subset of X . An element x ∈ (X ) is called a lower bound for A if x ≤ a, ∀a ∈ A. It is called the greatest lower bound if x is the largest lower bound and is denoted by glb(A) or in f (A).
Definition 1.1.20 LetA be a set and x, y ∈ A. We denote the supremum of the pair (x, y), called a join of x and y, by x y and the infimum, which is also called the meet, by x y.
Remark 1.1.21 A set which has both the supremum and the infimum is said to have both the join and the meet.
Remark 1.1.22 Let A be a set and a, b, c ∈ A. To say that c is the supremum of the pair {a, b}, we write c = a ∨ b and to say c is the infimum of the pair we write c = a ∧ b.
Definition 1.1.23 Let (X, ⊆) be a poset and x, y ∈ X with x y. An element x ∈ X is called an immediate predecessor of element y ∈ X if x < y and there does not exists z ∈ X such that x < z < y. y is an immediate successor of x if x < y and there does not exists z ∈ X such that x < z < y.
Definition 1.1.24 Let (X, ⊆) be a poset. An element x ∈ X is said to be maximal if there does not exist y ∈ X such that x < y and is said to be minimal if there does not exists a y X such that y < x.
Remark 1.1.25 From the above results, we note the following:
Let A be a nonempty set and a, b ∈ A, then
(i) The greatest lower bound (glb(A)) of A may not belong to A and this is also true for the least upper bound (lub(A)) of A.
(ii) An element may have more than one immediate predecessor or more than one immediate successor.
(iii) minimal or maximal elements of A belong to A but they are not necessarily unique.
(iv) The least element of A is a lower bound of A that also belong to A and the greatest element of A is an upper bound of A that also belong to A.
Definition 1.1.26 Let X be a non-empty set. A semi-laNice is a partial order(X, ≤) in which every pair of elements x, y ∈ X has a least upper bound say z (i.e z = x ∨ y for every x, y X ).
Definition 1.1.27 A laNice, L, is a partially ordered set in which every pair of elements x, y L has a least upper bound and a greatest lower bound.
Definition 1.1.28 A subspace laNice is a family of subspaces of H which is closed under the formation of arbitrary intersections and arbitrary linear spans and which contains the zero subspace 0 and H.
Remark 1.1.29 The subspace laNice of all invariant, reducing and hyperinvariant sub- spaces of T ∈ B(H) is denoted by Lat (T ), Red(T ) and HyperLat(T ) respectively.
Definition 1.1.30 A laNice L of subspaces of H is said to be trivial if L = {{0}, {H}}.
Definition 1.1.31 An operator T ∈ B(H) is said to be:
Self-adjoint or Hermitian if T ∗ = T
Unitary if T ∗T = TT ∗ = I
Normal if T ∗T = TT ∗ an isometry if T ∗T = I a co-isometry if TT ∗ = I a partial isometry if T = TT ∗T quasinormal if T (T ∗T ) = (T ∗T )T hyponormal if T ∗T ≥ TT ∗ le shi operator if Tx = y where x = (x1, x2 · · · ) and y = (x2, x3 · · · ) right shi operator if Tx = y, where x = (x1, x2 · · · ) and y = (0, x1, x2 · · · ).
1.2 Some Bounded Operators in Hilbert Spaces
We recall that an operator is a continuous linear transformation between normed spaces over the same field. In this section, we define some bounded operators in Hilbert spaces and show how they are related.
Definition 1.2.1 Let T ∈ B(H, K) be an operator. The adjoint of the operator T denoted by, T ∗ is the unique mapping of K into H, such that
< Tx, y >=< x, T ∗y > ∀x ∈ H, y ∈ K
We note that T ∗ is bounded.
Proposition 1.2.2 Let T, S ∈ B(H, K) be operators and λ ∈ C be a scalar. The following properties hold true in general:
(i) (S + T )∗ = S∗ + T ∗
(ii) (λ T )∗ = λ¯ T ∗
(iii) (ST )∗ = S∗T ∗
(iv) I∗ = I
(v) (T ∗)∗ = T
(vi) ǁT ∗T ǁ = ǁT ǁ2.
Definition 1.2.3 An operator A ∈ B(H) is said to be positive if A is self-adjoint and < Ax, x > ≥ 0 ∀x ∈ H.
We note that for a self adjoint operator A, then A is positive and for any operator A ∈ B(H), both A∗A and AA∗ are positive.
Proposition 1.2.4 If A, B ∈ B(H) are positive and for α ≥ 0, then A + B is positive and so is αA.
Definition 1.2.5 An operator P ∈ B(H) is said to be idempotent if P2 = P.
Remark 1.2.6 If P ∈ B(H) is idempotent, then Ran(P) = Ker(I − P)so that Ran(P) is a subspace of H.
Definition 1.2.7 An operator T ∈ B(H) is called a projection if T 2 = T .
Definition 1.2.8 An operator T ∈ B(H) is called an orthogonal projection if T is idem- potent and self-adjoint(i.e if T 2 = T and T ∗ = T ).
Remark 1.2.9 Both the projection and orthogonal projection operators are positive and self-adjoint.
Definition 1.2.10 An operator T ∈ B(H, K) is said to be invertible if it has an inverse and the range Ran(T ) = K, and such an inverse must be bounded.
Proposition 1.2.11 Let H be a Hilbert space and T ∈ B(H) be an operator. Then T is invertible if and only if T ∗ is invertible.
Proposition 1.2.13 A unitary operator is an invertible isometry.
Definition 1.2.14 An operator T ∈ B(H) is normal if it commutes with its adjoint (i.e T ∗T = TT ∗ or 0 = T ∗T − TT ∗).
Definition1.2.15 Let T ∈ B(H) be an operator and λ ∈ C. Then T is said to be hyponor- mal if TT ∗ ≤ T ∗T (i.e(λ I − T )(λ I − T )∗ ≤ (λ I − T )∗(λ I − T )).
Definition 1.2.16 An operatorT ∈ B(H) is said to be cohyponormal if its adjoint is hy- ponormal (i,e if T ∗T ≥ TT ∗) for λ ∈ C.
Remark 1.2.17 T ∈ B(H) is normal if and only if it is both hyponormal and cohyponor- mal.
Definition 1.2.18 An operator T ∈ B(H) is said to be semi-normal if T is either a hy- ponormal or cohyponormal operator.
Theorem 1.2.19 Let T be an operator. Then the following assertions are equivalent
(a) T is normal
(b) ǁT ∗xǁ = ǁTxǁ for every x ∈ H
(c) Tn is normal for every integer n ≥ 1
(d) T ∗nx = Tnx for every x H and every integer n 1.
Remark 1.2.20 Every self adjoint and unitary operator is normal.
Proposition 1.2.21 An operator P ∈ B(H) is an orthogonal projection if an only if it is a normal projection.
Definition 1.2.21 An operator T ∈ B(H, K) is compact if for every bounded sequence xn H, the sequence Txn K has a convergent subsequence .
Proposition 1.2.22 let T be a unitary operator on H. Then T is compact if and only if H has a finite dimension.
Definition 1.2.23 Two operators A, B ∈ B(H, K) are said to be similar if there exists an invertible operator S ∈ B(H, K) such that SA = BS or equivalently, A = S−1BS.
Definition 1.2.24[15] Two operators A, B ∈ B(H, K) are said to be almost similar if there exists an invertibe operator S ∈ B(H, K) such that A∗A = S−1(B∗B)S and A∗ + A = S−1(B∗ + B)S.
Definition 1.2.25 Two operators A, B ∈ B(H, K) are unitarily equivalent if there exists a unitary operator U ∈ B+(H, K)(Banach algebra of all invertible operators in B(H)) such that UA = BU (i.e A = U ∗BU , equivalently, A = U −1BU ).
Definition 1.2.26 Two operators A, B ∈ B(H) are said to be almost unitarily equiva- lent if there exists a unitary operator U ∈ B+(H, K) such that A∗A = U ∗(B∗B)U and A∗ + A = U ∗(B∗ + B)U.
Proposition 1.2.27 If A, B ∈ B(H) are unitarily equivalent, then they are almost similar.
Definition 1.2.28 An operator X ∈ B(H, K) is said to be a quasi- affinity or quasi-invertible if it is injective and has a dense range.
i,eN(T ) = {0} and R(T ) = K
Equivalently,N(T ) = {0} and N(T ∗) = 0.
Proposition 1.2.29 An operator T ∈ B(H) is quasi-invertible if and only if T ∗ is quasi- invertible.
Definition 1.2.30 Two operators A, B ∈ B(H) are said to be quasiaffine transforms of each other if there exists a quasi-affinity X B(H, K) such that AX = BX .
Definition 1.2.31 Two operators are said to be quasisimilar if there exists quasi-affinities
X ∈ B(H, K) and Y ∈ B(K, H) such that XA = BX and AY = YB.
Definition 1.2.32 Two operators A, B ∈ B(H, K) are said to be metrically equivalent if ǁAxǁ = ǁBxǁ or equivalently |< Ax, Ax >| 2 = |< Bx, Bx >| 2 .
Definition 1.2.33 Two operators A, B ∈ B(H, K) are said to be α−metrically equivalent if there exists an α > 0 such that A∗A = α2B∗B.
Definition 1.2.34 [31] An operator T ∈ B(H) is said to be quasi-unitary if T ∗T = TT ∗ = T ∗ + T .
Proposition 1.2.35 If T ∈ B(H) is quasi-unitary, then T ∗ is also quasi-unitary. Definition 1.2.36 Two operators A, B ∈ B(H) are said to be absolutely equivalent if both the absolute values of the operators are unitarily equivalent (i.e |A| = U |B|U ∗).
Definition 1.2.37[25] Two operators A, B ∈ B(H) are said to be nearly-equivalent if there exists unitary operator U ∈ B(H) such that A∗A = UB∗BU ∗.
Definition 1.2.38 An operator T ∈ B(H) is said to be reducible if it has nontrivial reduc- ing subspace.
Definition 1.2.39 An operator T ∈ B(H) is said to be reductive if all its invariant sub- spaces reduce it.
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