Linear Operators and Spectral Theory

238 Pages·2003·1.63 MB·english

Linear Operators and Spectral Theory
Applied Mathematics Seminar - V.I.
Math 488, Section 1, WS2003
Regular Participants:
V. Batchenko
V. Borovyk
R. Cascaval
D. Cramer
F. Gesztesy
O. Mesta
K. Shin
M. Zinchenko
Additional Participants:
C. Ahlbrandt
Y. Latushkin
K. Makarov
Coordinated by F. Gesztesy
1
Contents
V. Borovyk: Topics in the Theory of Linear Operators in Hilbert Spaces
O. Mesta: Von Neumann’s Theory of Self-Adjoint Extensions of Sym-
metric Operators and some of its Refinements due to Friedrichs and Krein
D. Cramer: Trace Ideals and (Modified) Fredholm Determinants
F. Gesztesy and K. A. Makarov: (Modified) Fredholm Determinants
for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited
M. Zinchenko: Spectral and Inverse Spectral Theory of Second-Order
Difference (Jacobi) Operators on N and on Z
K. Shin: Floquet and Spectral Theory for Second-Order Periodic Differ-
ential Equations
K. Shin: On Half-Line Spectra for a Class of Non-Self-Adjoint Hill Op-
erators
V. Batchenko and F. Gesztesy: On the Spectrum of Quasi-Periodic
Algebro-Geometric KdV Potentials
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Topics in the Theory of Linear Operators
in Hilbert Spaces
Vita Borovyk
Math 488, Section 1
Applied Math Seminar - V.I., WS 2003
February, 2003
- The spectral theorem for bounded and un-
bounded self-adjoint operators
- Characterizations of the spectrum , point spec-
trum, essential spectrum, and discrete spectrum
of a self-adjoint operator
- Stone’s theorem for unitary groups
- Singular values of compact operators, trace
class and Hilbert–Schmidt operators
1
1 Preliminaries
For simplicity we will always assume that the Hilbert spaces considered in
this manuscript are separable and complex (although most results extend to
nonseparable complex Hilbert spaces).
Let H , H be separable Hilbert spaces and A be a linear operator A :
1 2
D(A) ⊂ H → H .
1 2
We denote by B(H ,H ) the set of all bounded linear operators from H
1 2 1
into H and write B(H,H) = B(H) for simplicity.
2
We recall that A = B if D(A) = D(B) = D and Ax = Bx for all x ∈ D.
Next, let H = H = H.
1 2
Definition 1.1. (i) Let T be densely defined in H. Then T∗ is called the
adjoint of T if,
Dom(T∗) = {g ∈ H|there exists an h ∈ H such that (h ,f) = (g,Tf) for
g g
all f ∈ Dom(T)},
∗
T g = h .
g
(ii) An operator A in H is called symmetric if A is densely defined and
A ⊆ A∗.
(iii) A densely defined operator B in H is called self-adjoint if B = B∗.
(iv) A densely defined operator S in H is called normal if SS∗ = S∗S.
We note that for every self-adjoint operator A in H one has D(A) = H.
For every bounded operator A we will assume D(A) = H unless explicitly
stated otherwise.
Definition 1.2. (i) z ∈ C lies in the resolvent set of A if (A−zI)−1 exists
and is bounded. The resolvent set of A is denoted by ρ(A).
(ii) If z ∈ ρ(A), then (A−zI)−1 is called the resolvent of A at the point z.
(iii) σ(A) = C\ρ(A) is called the spectrum of A.
We will use the notation,
R(z,A) = (A−zI)−1, z ∈ ρ(A).
Fact 1.3. σ(A) = σ(A).
Fact 1.4. A = A∗ ⇒ σ(A) ⊆ R.
2
Fact 1.5. If A is a bounded operator, then σ(A) is a bounded subset of C.
Fact 1.6. IfAisaboundedself-adjointoperator, thenσ(A) ⊂ Riscompact.
Fact 1.7. If A is a bounded self-adjoint operator, then (cid:6)A(cid:6) = sup |λ|.
λ∈σ(A)
Fact 1.8. If A is a self-adjoint operator, then R(z,A) is a normal operator
for all z ∈ ρ(A).
2 The spectral theorem for bounded
self-adjoint operators
Let H be a separable Hilbert space and A = A∗ ∈ B(H). We recall that
σ(A) ⊂ R is compact in this case.
Theorem 2.1. ([3], Thm. VII.1; the continuous functional calculus.)
There is a unique map ϕ : C(σ(A)) → B(H) such that for all f,g ∈
A
C(σ(A)):
ϕ (fg) = ϕ (f)ϕ (g),
A A A
ϕ (λf) = λϕ (f),
(i) A A
ϕ (1) = I,
A
∗
ϕ (f) = ϕ (f) .
A A
(These four conditions mean that ϕ is an algebraic *-homomorphism).
A
(ii) ϕ (f +g) = ϕ (f)+ϕ (g) (linearity).
A A A
(iii) (cid:6)ϕ (f)(cid:6) ≤ C(cid:6)f(cid:6) (continuity).
A B(H) ∞
(iv) If f(x) = x, then ϕ (f) = A.
A
Moreover, ϕ has the following additional properties:
A
(v) If Aψ = λψ, then ϕ (f)ψ = f(λ)ψ.
A
(vi) σ(ϕ (f)) = f(σ(A)) = {f(λ)|λ ∈ σ(A)} (the spectral mapping theo-
A
rem).
(vii) If f ≥ 0, then ϕ (f) ≥ 0.
A
(viii) (cid:6)ϕ (f)(cid:6) = (cid:6)f(cid:6) (this strengthens (iii)).
A B(H) ∞
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In other words, ϕ (f) = f(A).
A
Proof. (i), (ii)and(iv)uniquelydetermineϕ (p)foranypolynomialp. Since
A
polynomials are dense in C(σ(A)) (by the Stone–Weierstrass theorem), one
only has to show that
(cid:6)p(A)(cid:6) ≤ C sup |p(λ)|. (2.1)
B(H)
λ∈σ(A)
Then ϕ can be uniquely extended to the whole C(σ(A)) with the same
A
bound and the first part of the theorem will be proven. Equation(2.1) follows
from the subsequent two lemmas.
Now (viii) is obvious and properties (v), (vi) and (vii) follow easily as
well.
Lemma 2.2. σ(p(A)) = p(σ(A)) = {p(λ)|λ ∈ σ(A)}.
Lemma 2.3. (cid:6)p(A)(cid:6) = sup |p(λ)|.
λ∈σ(A)
Proof. Using property (i), Fact 1.7, and Lemma 2.2, one gets
(cid:6)p(A)(cid:6)2 = (cid:6)p(A)∗p(A)(cid:6) = (cid:6)(pp)(A)(cid:6) = sup |λ|
λ∈σ((pp)(A))
(cid:1) (cid:2)
2
= sup |p(λ)| .
λ∈σ(A)
Since it is not sufficient to have a functional calculus only for continuous
functions (the main goal of this construction is to define spectral projections
of the operator A which are characteristic functions of A), we have to extend
it to the space of bounded Borel functions, denoted by Bor(R).
Definition 2.4. f ∈ Bor(R) if f is a measurable function with respect to
the Borel measure on R and sup |f(x)| < ∞.
x∈R
Theorem 2.5. ([3], Thm. VII.2.)
Let A = A∗ ∈ B(H). Then there is a unique map ϕ(cid:3) : Bor(R) → B(H) such
A
that for all f,g ∈ Bor(R) the following statements hold:
(i) ϕ(cid:3) is an algebraic *-homomorphism.
A
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(ii) ϕ(cid:3) (f +g) = ϕ(cid:3) (f)+ϕ(cid:3) (g) (linearity).
A A A
(iii) (cid:6)ϕ(cid:3) (f)(cid:6) ≤ (cid:6)f(cid:6) (continuity).
A B(H) ∞
(iv) If f(x) = x, then ϕ(cid:3) (f) = A.
A
(v) If f (x) → f(x) for all x ∈ R, and f (x) are uniformly bounded w.r.t.
n n
n→∞
(x,n), then ϕ(cid:3) (f ) → ϕ(cid:3) (f) strongly.
A n A
n→∞
Moreover, ϕ(cid:3) has the following additional properties:
A
(vi) If Aψ = λψ, then ϕ(cid:3) (f)ψ = f(λ)ψ.
A
(vii) If f ≥ 0, then ϕ(cid:3) (f) ≥ 0.
A
(viii) If BA = AB, then Bϕ(cid:3) (f) = ϕ(cid:3) (f)B.
A A
Again, formally, ϕ(cid:3) (f) = f(A).
A
Proof. This theorem can be proven by extending the previous theorem. (One
has to invoke that the closure of C(R) under the limits of the form (v) is
precisely Bor(R).)
3 Spectral projections
Let BR denote the set of all Borel subsets of R.
Definition 3.1. The family {PΩ}Ω∈BR of bounded operators in H is called
a projection-valued measure (p.v.m.) of bounded support if the following con-
ditions (i)–(iv) hold:
(i) PΩ is an orthogonal projection for all Ω ∈ BR.
(ii) P = 0, there exist a,b ∈ R, a < b such that P = I (the bounded
∅ (a,b)
support property).
(cid:4)
(iii) If Ω = ∪∞k=1Ωk, Ωi∩Ωj = ∅ for i (cid:13)= j, then PΩ = s−limN→∞ Nk=1PΩk.
(iv) PΩ1PΩ2 = PΩ1∩Ω2.
Next, let A = A∗ ∈ B(H), Ω ∈ BR.
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Definition 3.2. P (A) = χ (A) are called the spectral projections of A.
Ω Ω
We note that the family {PΩ(A) = χΩ(A)}Ω∈BR satisfies conditions (i)–
(iv) of Definition 3.1.
Next, consider a p.v.m. {PΩ}Ω∈BR. Then for any h ∈ H, (h,PΩh) is
a positive (scalar) measure since properties (i)–(iv) imply all the necessary
properties of a positive measure. We will use the symbol d(h,P h) to denote
λ
the integration with respect to this measure.
By construction, the support of every (h,P (A)h) is a subset of σ(A).
Ω
Hence, if we integrate with respect to the measure (h,P h), we integrate
Ω
over σ(A). If we are dealing with an arbitrary p.v.m. we will denote the
support of the corresponding measure by supp(P ).
Ω
Theorem 3.3. ([3], Thm. VII.7.)
If {PΩ}Ω∈BR is a p.v.m. and f is a bounded Borel func(cid:5)tion on supp(PΩ), then
there is a unique operator B, which we will denote by f(λ)dP , such
supp(PΩ) λ
that
(cid:6)
(h,Bh) = f(λ)d(h,P h), h ∈ H. (3.1)
λ
supp(PΩ)
Proof. A standard Riesz argument.
Next, we will show that if P (A) is a p.v.m. associated with A, then
Ω
(cid:6)
f(A) = f(λ)dP (A). (3.2)
λ
σ(A)
First, assume f(λ) = χ (λ). Then
Ω
(cid:6) (cid:6)
χ (λ)d(h,P (A)h) = d(h,P (A)h) = (h,P (A)h)
Ω λ λ Ω
σ(A) σ(A)∩Ω
= (h,χ (A)h).
Ω
Hence, (3.2) holds for all simple functions. Next, approximate any mea-
surable function f(λ) by a sequence of simple functions to obtain (3.2) for
bounded Borel functions on σ(A).
The inverse statement(cid:5)also holds: If we start from any bounded p.v.m.
{PΩ}Ω∈BR and form A = supp(PΩ)λdPλ, then χΩ(A) = PΩ(A) = PΩ. This
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(cid:5)
follows from the fact that for such an A, the mapping f (cid:14)→ f(λ)dP
supp(PΩ) λ
forms a functional calculus for A. By uniqueness of the functional calculus
one then gets
(cid:6)
P (A) = χ (A) = χ (λ)dP = P .
Ω Ω Ω λ Ω
supp(PΩ)
Summarizing, one obtains the following result:
Theorem 3.4. ([3], Thm. VII.8; the spectral theorem in p.v.m. form.)
There is a one-to-one correspondence between bounded self-adjoint operators
A and projection-valued measures {PΩ}Ω∈BR in H of bounded support given
by
A → {PΩ(A)}Ω∈BR =(cid:6) {χΩ(A)}Ω∈BR,
{PΩ}Ω∈BR → A = λdPλ.
supp(PΩ)
4 The spectral theorem for unbounded self-
adjoint operators
The construction of the spectral decomposition for unbounded self-adjoint
operators will be based on the following theorem.
Theorem 4.1. ([3], Thm. VIII.4.)
∗
Assume A = A . Then there is a measure space (M ,dµ ) with µ a fi-
A A A
nite measure, a unitary operator U : H → L2(M ,dµ ), and a real-valued
A A A
function f on M which is finite a.e., such that
A A
(i) ψ ∈ D(A) ⇔ f (·)(U ψ)(·) ∈ L2(M ,dµ ).
A A A A
(ii) If ϕ ∈ U[D(A)], then (U AU−1ϕ)(m) = f (m)ϕ(m).
A A A
To prove this theorem we need some additional constructions. First we
will prove a similar result for bounded normal operators.
Definition 4.2. Let A be a bounded normal operator in H. Then ψ ∈ H is
a star-cyclic vector for A if
Lin.span{An(A∗)mψ}n,m∈N0 = H.
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Lemma 4.3. Let A be a bounded normal operator in H with a star-cyclic
vector ψ ∈ H. Then there is a measure µ on σ(A), and a unitary operator
A
U , such that U : H → L2(σ(A),dµ ) with
A A A
(U AU−1f)(λ) = λf(λ).
A A
This equality holds in the sense of equality of elements of L2(σ(A),dµ ).
A
(cid:4)
Proof. Introduce P = { n c λiλj,c ∈ C,n ∈ N} and take any p(·) ∈ P.
i,j=0 ij ij
Define U by U p(A)ψ = p. One can prove that for all x,y ∈ H there exists
A A
a measure µ on σ(A) such that
x,y,A
(cid:6)
(p(A)x,y) = p(λ)dµ , p ∈ P.
x,y,A
σ(A)
Then
(cid:6)
(cid:6)p(A)(cid:6)2 = (p(A)∗p(A)ψ,ψ) = ((pp)(A)ψ,ψ) = p(λ)p(λ)dµ
ψ,ψ,A
σ(A)
= (cid:6)p(cid:6)2 . (4.1)
L2(σ(A),dµψ,ψ,A)
Next we choose µ = µ . Since ψ is star-cyclic, U is densely defined and
A ψ,ψ,A A
equation (4.1) implies that U is bounded. Thus, U can be extended to an
A A
isometry
U : H → L2(σ(A),dµ ).
A A
Since P(σ(A)) is dense in L2(σ(A),dµ ), Ran(U ) = L2(σ(A),dµ ) and U
A A A A
is invertible. Thus, U is unitary.
A
Finally, if p ∈ P(σ(A)), then
(U AU−1p)(λ) = (U Ap(A)ψ)(λ) = (U (λ·p)(A)ψ)(λ) = λp(λ).
A A A A
By continuity, this can be extended from P(σ(A)) to L2(σ(A),dµ ).
A
Lemma 4.4. Let A be a bounded normal operator on a separable Hilbert
space H. Then there is an orthogonal direct sum decomposition H = ⊕N H
j=1 j
(N ≤ ∞) such that:
(i) For all j: AH ⊆ H .
j j
(ii) For all j there exists an xj ∈ Hj such that xj is star-cyclic for A|Hj.
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