Isolated eigenvalues of linear

operator and perturbations

Slavisa DjordjevicBenemerita Universidad Autonoma de Puebla

WCAOS 2010, September 2–4.

Basic notation

WCAOS 2010 S. Djordjevic – 2 / 18

■ Throughout this talk let X be a Banach space and B(X) be the set ofbounded linear operators acting on X.

■ For T ∈ B(X) the spectrum of T is defined by

σ(T ) = {λ ∈ C : T − λI is not invertible}

and the resolvent setρ(T ) = C \ σ(T ).

■ The complex number λ is called an eigenvalue of T if exists a non-zerovector x ∈ X such that Tx = λx (or equivalent (T − λI)x = 0). Theset of all eigenvalues of T we denote σp(T ).

Basic notation

WCAOS 2010 S. Djordjevic – 2 / 18

■ Throughout this talk let X be a Banach space and B(X) be the set ofbounded linear operators acting on X.

■ For T ∈ B(X) the spectrum of T is defined by

σ(T ) = {λ ∈ C : T − λI is not invertible}

and the resolvent setρ(T ) = C \ σ(T ).

■ The complex number λ is called an eigenvalue of T if exists a non-zerovector x ∈ X such that Tx = λx (or equivalent (T − λI)x = 0). Theset of all eigenvalues of T we denote σp(T ).

Eigenvalues

WCAOS 2010 S. Djordjevic – 3 / 18

■ Λ ⊂ σ(T ) is called spectral set for T if both Λ and σ(T ) \ Λ are closedin relative topology of σ(T ).

■ For a spectral set Λ of T with C(T, Λ) we denote the set of all Cauchycontour C which separate Λ from σ(T ) \ Λ.

■ For z ∈ ρ(T ),R(T, z) = (T − zI)−1

is called the resolvent operator of T at z.

Eigenvalues

WCAOS 2010 S. Djordjevic – 3 / 18

■ Λ ⊂ σ(T ) is called spectral set for T if both Λ and σ(T ) \ Λ are closedin relative topology of σ(T ).

■ For a spectral set Λ of T with C(T, Λ) we denote the set of all Cauchycontour C which separate Λ from σ(T ) \ Λ.

■ For z ∈ ρ(T ),R(T, z) = (T − zI)−1

is called the resolvent operator of T at z.

Eigenvalues

WCAOS 2010 S. Djordjevic – 3 / 18

■ Λ ⊂ σ(T ) is called spectral set for T if both Λ and σ(T ) \ Λ are closedin relative topology of σ(T ).

■ For a spectral set Λ of T with C(T, Λ) we denote the set of all Cauchycontour C which separate Λ from σ(T ) \ Λ.

■ For z ∈ ρ(T ),R(T, z) = (T − zI)−1

is called the resolvent operator of T at z.

Spectral sets

WCAOS 2010 S. Djordjevic – 4 / 18

■ For a spectral set Λ for T ∈ B(X) and C ∈ C(T, Λ), define

P (T, Λ) = −1

2πi

∫

C

R(T, z)dz

a bounded projection of T and Λ.

■ For λ ∈ σ(T ) we say that is Riesz point of T if λ is an isolatedeigenvalues of T of finite algebraic multiplicity (ordimP (T, λ)(X) < ∞).

■ For an isolated eigenvalue λ of T we say that has finite geometricmultiplicity if dimN(T − λI) < ∞.

■ Let π0(T ) denote the set of Riesz points of T and let π00(T ) denote theset of eigenvalues of T of finite geometric multiplicity.

Spectral sets

WCAOS 2010 S. Djordjevic – 4 / 18

■ For a spectral set Λ for T ∈ B(X) and C ∈ C(T, Λ), define

P (T, Λ) = −1

2πi

∫

C

R(T, z)dz

a bounded projection of T and Λ.

■ For λ ∈ σ(T ) we say that is Riesz point of T if λ is an isolatedeigenvalues of T of finite algebraic multiplicity (ordimP (T, λ)(X) < ∞).

■ For an isolated eigenvalue λ of T we say that has finite geometricmultiplicity if dimN(T − λI) < ∞.

■ Let π0(T ) denote the set of Riesz points of T and let π00(T ) denote theset of eigenvalues of T of finite geometric multiplicity.

Spectral sets

WCAOS 2010 S. Djordjevic – 4 / 18

■ For a spectral set Λ for T ∈ B(X) and C ∈ C(T, Λ), define

P (T, Λ) = −1

2πi

∫

C

R(T, z)dz

a bounded projection of T and Λ.

■ For λ ∈ σ(T ) we say that is Riesz point of T if λ is an isolatedeigenvalues of T of finite algebraic multiplicity (ordimP (T, λ)(X) < ∞).

■ For an isolated eigenvalue λ of T we say that has finite geometricmultiplicity if dimN(T − λI) < ∞.

■ Let π0(T ) denote the set of Riesz points of T and let π00(T ) denote theset of eigenvalues of T of finite geometric multiplicity.

Spectral sets

WCAOS 2010 S. Djordjevic – 4 / 18

■ For a spectral set Λ for T ∈ B(X) and C ∈ C(T, Λ), define

P (T, Λ) = −1

2πi

∫

C

R(T, z)dz

a bounded projection of T and Λ.

■ For λ ∈ σ(T ) we say that is Riesz point of T if λ is an isolatedeigenvalues of T of finite algebraic multiplicity (ordimP (T, λ)(X) < ∞).

■ For an isolated eigenvalue λ of T we say that has finite geometricmultiplicity if dimN(T − λI) < ∞.

■ Let π0(T ) denote the set of Riesz points of T and let π00(T ) denote theset of eigenvalues of T of finite geometric multiplicity.

Spectral sets

WCAOS 2010 S. Djordjevic – 5 / 18

■ It is known that π0(T ) ⊂ π00(T )

■ An eigenvalue λ ∈ π0(T ) is called simple eigenvalues if dimP (T, λ) = 1,or equivalent X = N (T − λI) ⊕R(T − λI).

Spectral sets

WCAOS 2010 S. Djordjevic – 5 / 18

■ It is known that π0(T ) ⊂ π00(T )

■ An eigenvalue λ ∈ π0(T ) is called simple eigenvalues if dimP (T, λ) = 1,or equivalent X = N (T − λI) ⊕R(T − λI).

Motivation: Approximation of simple

eigenvalues for bounded operators

WCAOS 2010 S. Djordjevic – 6 / 18

■ Remark. If λ ∈ π0(T ), then there exists a sequence λn ∈ π0(Tn) suchthat λn → λ and dimP (T, λ) = dimP (Tn, λn).

Refinement schemes for a simple eigenvalue

■ Theorem. Let λ be a simple eigenvalue of T and φ be a correspondingeigenvector. Assume that Tn −→ T .

Then for each large enough n, Tn has a unique simple eigenvalue λn

such that λn → λ.

Motivation: Approximation of simple

eigenvalues for bounded operators

WCAOS 2010 S. Djordjevic – 6 / 18

■ Remark. If λ ∈ π0(T ), then there exists a sequence λn ∈ π0(Tn) suchthat λn → λ and dimP (T, λ) = dimP (Tn, λn).

Refinement schemes for a simple eigenvalue

■ Theorem. Let λ be a simple eigenvalue of T and φ be a correspondingeigenvector. Assume that Tn −→ T .

Then for each large enough n, Tn has a unique simple eigenvalue λn

such that λn → λ.

Motivation: Approximation of simple

eigenvalues for bounded operators

WCAOS 2010 S. Djordjevic – 7 / 18

Let φn be an eigenvector of Tn corresponding to λn and φ∗

n be theeigenvector of T ∗

n corresponding to its eigenvalue λn such that 〈φn, φ∗

n〉 = 1.Then 〈φ, φ∗

n〉 6= 0 for all large n. If we let

φ(n) =φ

〈φ, φ∗

n〉

then for all large n, we have

max

{

|λn − λ|,‖φn − φ(n)‖

‖φn‖

}

≤ c‖Tn − T‖

and if λ 6= 0, then

max

{

|λn − λ|,‖φn − φ(n)‖

‖φn‖

}

≤ c‖(Tn − T )T‖,

where c is a constant, independent of n.

Examples

WCAOS 2010 S. Djordjevic – 8 / 18

• M. Ahues, A. Largiller and B.V. Limaye, Spectral Computations forBounded Operators, Chapman-Hall/CRC, 2001

Finite rank approximations

Let X be a complex Banach space and T a bounded liner operator. Withsome extra conditions, for example if T is a compact operator, or X hasSchauder basis, we can find {Tn} a sequence of finite rank operators suchthat Tn converge in norm to T . Since rank of operators Tn are finite, then thespectral computations for Tn can be reduced to solving a matrix eigenvalueproblem in a canonical way. For this reason we will present various situationswhen we can apply this technics.

Examples

WCAOS 2010 S. Djordjevic – 9 / 18

Approximation based on projections

Let (πn) be a sequence of bounded linear projection defined on a Banachspace X. Define:

TPn = πnT, TS

n = Tπn and TGn = πnTπn.

The bounded operators TPn , TP

n and TPn are known as the projection

approximation of T , Sloan approximation of T and Galerkin approximation ofT , respectively.Theorem. Let T ∈ B(X) and πn(x) −→ x(= I(x)). Then• If T is compact operator, then TP

n −→ T ;• If T is compact operator and π∗

n(·) −→ I∗(·), then TSn −→ T and

TGn −→ T .

Examples

WCAOS 2010 S. Djordjevic – 10 / 18

Truncation of a Schauder expansion

Assume that X has a Schauder basis (ei). For each positive integer n define

πn(x) =∑n

j=1 cj(x)ej , x ∈ X.

Then for T ∈ B(X) such that Tej =∑

∞

j=1 ti,jei, j = 1, 2, . . . we have

TPn ej =

∑nj=1 ti,jei, j = 1, 2, . . .

TSn ej =

{ ∑

∞

j=1 ti,jei, j = 1, 2, . . . , n

0, j > n.

TGn ej =

{ ∑nj=1 ti,jei, j = 1, 2, . . . , n

0, j > n.

Localization of eigenvalues of linear

operators

WCAOS 2010 S. Djordjevic – 11 / 18

■ K(X) denotes the ideal of all compact operators

■ α(T ) = dimN(T ); β(T ) = dim(X/R(T ))

■ φ+(X) = {T ∈ B(X) : R(T ) is closed and α(T ) < ∞}

φ−(X) = {T ∈ B(X) : β(T ) < ∞}

■ An operator T ∈ B(H) is called semi-Fredholm if T ∈ φ+(X) ∪ φ−(X)

An operator T ∈ B(H) is called Fredholm if T ∈ φ+(X) ∩ φ−(X)

■ The index of T ∈ φ+(X) ∪ φ−(X) is given by ind(T ) = α(T ) − β(T ).

Localization of eigenvalues of linear

operators

WCAOS 2010 S. Djordjevic – 11 / 18

■ K(X) denotes the ideal of all compact operators

■ α(T ) = dimN(T ); β(T ) = dim(X/R(T ))

■ φ+(X) = {T ∈ B(X) : R(T ) is closed and α(T ) < ∞}

φ−(X) = {T ∈ B(X) : β(T ) < ∞}

■ An operator T ∈ B(H) is called semi-Fredholm if T ∈ φ+(X) ∪ φ−(X)

An operator T ∈ B(H) is called Fredholm if T ∈ φ+(X) ∩ φ−(X)

■ The index of T ∈ φ+(X) ∪ φ−(X) is given by ind(T ) = α(T ) − β(T ).

Localization of eigenvalues of linear

operators

WCAOS 2010 S. Djordjevic – 11 / 18

■ K(X) denotes the ideal of all compact operators

■ α(T ) = dimN(T ); β(T ) = dim(X/R(T ))

■ φ+(X) = {T ∈ B(X) : R(T ) is closed and α(T ) < ∞}

φ−(X) = {T ∈ B(X) : β(T ) < ∞}

■ An operator T ∈ B(H) is called semi-Fredholm if T ∈ φ+(X) ∪ φ−(X)

An operator T ∈ B(H) is called Fredholm if T ∈ φ+(X) ∩ φ−(X)

■ The index of T ∈ φ+(X) ∪ φ−(X) is given by ind(T ) = α(T ) − β(T ).

Localization of eigenvalues of linear

operators

WCAOS 2010 S. Djordjevic – 11 / 18

■ K(X) denotes the ideal of all compact operators

■ α(T ) = dimN(T ); β(T ) = dim(X/R(T ))

■ φ+(X) = {T ∈ B(X) : R(T ) is closed and α(T ) < ∞}

φ−(X) = {T ∈ B(X) : β(T ) < ∞}

■ An operator T ∈ B(H) is called semi-Fredholm if T ∈ φ+(X) ∪ φ−(X)

An operator T ∈ B(H) is called Fredholm if T ∈ φ+(X) ∩ φ−(X)

■ The index of T ∈ φ+(X) ∪ φ−(X) is given by ind(T ) = α(T ) − β(T ).

Localization of eigenvalues of linear

operators

WCAOS 2010 S. Djordjevic – 11 / 18

■ K(X) denotes the ideal of all compact operators

■ α(T ) = dimN(T ); β(T ) = dim(X/R(T ))

■ φ+(X) = {T ∈ B(X) : R(T ) is closed and α(T ) < ∞}

φ−(X) = {T ∈ B(X) : β(T ) < ∞}

■ An operator T ∈ B(H) is called semi-Fredholm if T ∈ φ+(X) ∪ φ−(X)

An operator T ∈ B(H) is called Fredholm if T ∈ φ+(X) ∩ φ−(X)

■ The index of T ∈ φ+(X) ∪ φ−(X) is given by ind(T ) = α(T ) − β(T ).

Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 12 / 18

■ φ0(X) = {T ∈ B(X) : T ∈ φ(X) and ind(T ) = 0} – Weyl operators

■ Weyl spectrum by σw(A) = {λ ∈ C : T − λ /∈ φ0(X)}

σw(T ) = ∩{σ(T + K) : K ∈ K(X)}

■ H. Weyl, Uber beschrankte quadratische formen, deren Differenzvollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

■ λ belongs to the spectra of all compact perturbations T + K of a singlehermitian operator T if and only if λ is not an isolated eigenvalue offinite multiplicity

■ We say that T obeys Weyl’s theorem if σw(T ) = σ(T ) \ π00(T )

Browder’s theorem if σw(T ) = σ(T ) \ π0(T )

Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 12 / 18

■ φ0(X) = {T ∈ B(X) : T ∈ φ(X) and ind(T ) = 0} – Weyl operators

■ Weyl spectrum by σw(A) = {λ ∈ C : T − λ /∈ φ0(X)}

σw(T ) = ∩{σ(T + K) : K ∈ K(X)}

■ H. Weyl, Uber beschrankte quadratische formen, deren Differenzvollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

■ λ belongs to the spectra of all compact perturbations T + K of a singlehermitian operator T if and only if λ is not an isolated eigenvalue offinite multiplicity

■ We say that T obeys Weyl’s theorem if σw(T ) = σ(T ) \ π00(T )

Browder’s theorem if σw(T ) = σ(T ) \ π0(T )

Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 12 / 18

■ φ0(X) = {T ∈ B(X) : T ∈ φ(X) and ind(T ) = 0} – Weyl operators

■ Weyl spectrum by σw(A) = {λ ∈ C : T − λ /∈ φ0(X)}

σw(T ) = ∩{σ(T + K) : K ∈ K(X)}

■ H. Weyl, Uber beschrankte quadratische formen, deren Differenzvollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

■ λ belongs to the spectra of all compact perturbations T + K of a singlehermitian operator T if and only if λ is not an isolated eigenvalue offinite multiplicity

■ We say that T obeys Weyl’s theorem if σw(T ) = σ(T ) \ π00(T )

Browder’s theorem if σw(T ) = σ(T ) \ π0(T )

Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 12 / 18

■ φ0(X) = {T ∈ B(X) : T ∈ φ(X) and ind(T ) = 0} – Weyl operators

■ Weyl spectrum by σw(A) = {λ ∈ C : T − λ /∈ φ0(X)}

σw(T ) = ∩{σ(T + K) : K ∈ K(X)}

■ H. Weyl, Uber beschrankte quadratische formen, deren Differenzvollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

■ λ belongs to the spectra of all compact perturbations T + K of a singlehermitian operator T if and only if λ is not an isolated eigenvalue offinite multiplicity

■ We say that T obeys Weyl’s theorem if σw(T ) = σ(T ) \ π00(T )

Browder’s theorem if σw(T ) = σ(T ) \ π0(T )

Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 12 / 18

■ φ0(X) = {T ∈ B(X) : T ∈ φ(X) and ind(T ) = 0} – Weyl operators

■ Weyl spectrum by σw(A) = {λ ∈ C : T − λ /∈ φ0(X)}

σw(T ) = ∩{σ(T + K) : K ∈ K(X)}

■ H. Weyl, Uber beschrankte quadratische formen, deren Differenzvollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

■ λ belongs to the spectra of all compact perturbations T + K of a singlehermitian operator T if and only if λ is not an isolated eigenvalue offinite multiplicity

■ We say that T obeys Weyl’s theorem if σw(T ) = σ(T ) \ π00(T )

Browder’s theorem if σw(T ) = σ(T ) \ π0(T )

Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 12 / 18

■ φ0(X) = {T ∈ B(X) : T ∈ φ(X) and ind(T ) = 0} – Weyl operators

■ Weyl spectrum by σw(A) = {λ ∈ C : T − λ /∈ φ0(X)}

σw(T ) = ∩{σ(T + K) : K ∈ K(X)}

■ H. Weyl, Uber beschrankte quadratische formen, deren Differenzvollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

■ λ belongs to the spectra of all compact perturbations T + K of a singlehermitian operator T if and only if λ is not an isolated eigenvalue offinite multiplicity

■ We say that T obeys Weyl’s theorem if σw(T ) = σ(T ) \ π00(T )

Browder’s theorem if σw(T ) = σ(T ) \ π0(T )

Some classes of operators

WCAOS 2010 S. Djordjevic – 13 / 18

Let H be a Hilbert space. An operator T ∈ B(H) is

■ normal if TT ∗ = T ∗T , or ‖T ∗x‖ = ‖Tx‖, for each x ∈ H,

■ T ∈ B(H) is hyponormal if TT ∗ ≤ T ∗T , or ‖T ∗x‖ ≤ ‖Tx‖, for eachx ∈ H,

■ T ∈ B(H) is p-hyponormal if ((T ∗T )p − TT ∗)p ≥ 0 holds,

■ T is quasi-hyponormal if ‖T ∗Tx‖ ≤ ‖T 2x‖, for each x ∈ H.

■ Theorem. Let Hilbert space operators T or T ∗ are in one of the classesabove. Then Weyl’s theorem holds for T (π00(T ) = π0(T )).

Some classes of operators

WCAOS 2010 S. Djordjevic – 13 / 18

Let H be a Hilbert space. An operator T ∈ B(H) is

■ normal if TT ∗ = T ∗T , or ‖T ∗x‖ = ‖Tx‖, for each x ∈ H,

■ T ∈ B(H) is hyponormal if TT ∗ ≤ T ∗T , or ‖T ∗x‖ ≤ ‖Tx‖, for eachx ∈ H,

■ T ∈ B(H) is p-hyponormal if ((T ∗T )p − TT ∗)p ≥ 0 holds,

■ T is quasi-hyponormal if ‖T ∗Tx‖ ≤ ‖T 2x‖, for each x ∈ H.

■ Theorem. Let Hilbert space operators T or T ∗ are in one of the classesabove. Then Weyl’s theorem holds for T (π00(T ) = π0(T )).

Perturbations and Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 14 / 18

■ Weyl‘s theorem for T ?=⇒? Weyl‘s theorem for T + K

Perturbations and Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 14 / 18

■ Weyl‘s theorem for T ?=⇒? Weyl‘s theorem for T + K

■ K.K. Oberai, On the Weyl‘s spectrum II, Illinois Math. J. 21 (1977),84–90.

■ M. Oudghiri, Weyl‘s theorem and Perturbations, Integr.Equ.Op.Theory53 (2005), 535–545.

■ P.Aiena, J.R. Guillen and P. Pena, Weyl‘s type theorems andPerturbations, Divulgaciones Mate. 16 (2008), 55–72.

Perturbations and Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 15 / 18

Theorem Weyl’s theorem is transmitted from T ∈ B(X) to T + N , when Nis nilpotent operator commuting with T .

Perturbations and Weyl‘s theorem

WCAOS 2010 S. Djordjevic – 15 / 18

Theorem Weyl’s theorem is transmitted from T ∈ B(X) to T + N , when Nis nilpotent operator commuting with T .

Example Let X = ℓ2(N) and T and N be defined by:

T (x1, x2, . . .) = (0, x1

2 , x2

3 , . . .)

N(x1, x2, . . .) = (0,−x1

2 , 0, . . .).

Then T obeys Weyl‘s theorem, but T + N not (T and N are notcommuting).

Compact Perturbations and Weyl‘s

theorem

WCAOS 2010 S. Djordjevic – 16 / 18

Note In the previous example N is also a finite rank operator not commutingwith T .

Compact Perturbations and Weyl‘s

theorem

WCAOS 2010 S. Djordjevic – 16 / 18

Note In the previous example N is also a finite rank operator not commutingwith T .

Example In general, Weyl‘s theorem is also not transmitted undercommuting finite rank perturbation.

Let X = ℓ2(N) and S ∈ B(X) be an injective quasi-nilpotent operatorand U be defined:

U(x1, x2, . . .) = (−x1, 0, . . .).

Define on Y = X ⊕ X the operators T and K by

T = ⊕S and K = U ⊕ O.

K is a finite rank operator commuting with T , T obeys Weyl‘s theorembut T + K not.

Compact Perturbations and Weyl‘s

theorem

WCAOS 2010 S. Djordjevic – 17 / 18

■ P. Aiena, J.R. Guillen, Weyl‘s theorem for perturbations of paranormaloperator, PAMS (2005)

Theorem Suppose that T ∈ B(H) is paranormal, K algebraic andTK = KT . Then Weyl‘s theorem is transmitted on T + K.

Compact Perturbations and Weyl‘s

theorem

WCAOS 2010 S. Djordjevic – 17 / 18

■ P. Aiena, J.R. Guillen, Weyl‘s theorem for perturbations of paranormaloperator, PAMS (2005)

Theorem Suppose that T ∈ B(H) is paranormal, K algebraic andTK = KT . Then Weyl‘s theorem is transmitted on T + K.

T is paranormal if ‖Tx‖2 ≤ ‖T 2x‖ · ‖x‖.

K is algebraic if exists a polynomial p such that p(K) = 0.

Compact Perturbations and Weyl‘s

theorem

WCAOS 2010 S. Djordjevic – 17 / 18

■ P. Aiena, J.R. Guillen, Weyl‘s theorem for perturbations of paranormaloperator, PAMS (2005)

Theorem Suppose that T ∈ B(H) is paranormal, K algebraic andTK = KT . Then Weyl‘s theorem is transmitted on T + K.

T is paranormal if ‖Tx‖2 ≤ ‖T 2x‖ · ‖x‖.

K is algebraic if exists a polynomial p such that p(K) = 0.

Theorem Suppose that T ∈ B(H) satisfies Weyl‘s theorem. If σ(T ) has noholes and has at most finitely many isolated points then Weyl‘s theoremholds for T + K for every compact operator K.

WCAOS 2010 S. Djordjevic – 18 / 18

.

Thank you.