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Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Borbala Mercedes Gerhat

Vienna University of Technology

December 20, 2016

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Perron-Frobenius Theorem

For a positive matrix A ∈ Rn×n, i.e. aij > 0, the following hold true:

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Perron-Frobenius Theorem

For a positive matrix A ∈ Rn×n, i.e. aij > 0, the following hold true:

• r(A) > 0 and r(A) ∈ σ(A) is algebraically simple, i.e.

geomA(r(A)) = algA(r(A)) = 1.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Perron-Frobenius Theorem

For a positive matrix A ∈ Rn×n, i.e. aij > 0, the following hold true:

• r(A) > 0 and r(A) ∈ σ(A) is algebraically simple, i.e.

geomA(r(A)) = algA(r(A)) = 1.

• There exists a positive eigenvector associated to r(A),

Ax = r(A)x with xi > 0.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Perron-Frobenius Theorem

For a positive matrix A ∈ Rn×n, i.e. aij > 0, the following hold true:

• r(A) > 0 and r(A) ∈ σ(A) is algebraically simple, i.e.

geomA(r(A)) = algA(r(A)) = 1.

• There exists a positive eigenvector associated to r(A),

Ax = r(A)x with xi > 0.

• Except the positive multiples of x, there are no other non-negative

eigenvectors of A.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Perron-Frobenius Theorem

For a positive matrix A ∈ Rn×n, i.e. aij > 0, the following hold true:

• r(A) > 0 and r(A) ∈ σ(A) is algebraically simple, i.e.

geomA(r(A)) = algA(r(A)) = 1.

• There exists a positive eigenvector associated to r(A),

Ax = r(A)x with xi > 0.

• Except the positive multiples of x, there are no other non-negative

eigenvectors of A.

• |λ| < r(A) for all λ ∈ r(A)\{r(A)}.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

A subset K of a real Banach space X is called an order cone, if (i) K is closed, K = ∅ and K = {0} (ii) K + K ⊆ K and αK ⊆ K for α ≥ 0 (iii) K ∩ (−K) = {0} The pair X, K is called an ordered Banach space.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

A subset K of a real Banach space X is called an order cone, if (i) K is closed, K = ∅ and K = {0} (ii) K + K ⊆ K and αK ⊆ K for α ≥ 0 (iii) K ∩ (−K) = {0} The pair X, K is called an ordered Banach space.

• In (ii), requirement K + K ⊆ K can be replaced by convexity of K.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

A subset K of a real Banach space X is called an order cone, if (i) K is closed, K = ∅ and K = {0} (ii) K + K ⊆ K and αK ⊆ K for α ≥ 0 (iii) K ∩ (−K) = {0} The pair X, K is called an ordered Banach space.

• In (ii), requirement K + K ⊆ K can be replaced by convexity of K.
• 0 ∈ K, but 0 /

∈ K ◦: (Ur(0) ⊆ K, r > 0) ±

r 2xx ∈ Ur(0) ⊆ K,

x ∈ X\{0} arbitrary.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

Let X, K be an ordered Banach space. The order cone K is called generating, if span(K) = X and total, if span(K) = X.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

Let X, K be an ordered Banach space. The order cone K is called generating, if span(K) = X and total, if span(K) = X.

• K is generating (resp. total), if and only if X = K − K (resp.

X = K − K): span(K) ∋

n

• i=1

aixi =

• ai≥0

aixi −

• ai<0

(−ai)xi ∈ K − K

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

Let X, K be an ordered Banach space. The order cone K is called generating, if span(K) = X and total, if span(K) = X.

• K is generating (resp. total), if and only if X = K − K (resp.

X = K − K): span(K) ∋

n

• i=1

aixi =

• ai≥0

aixi −

• ai<0

(−ai)xi ∈ K − K

• If K ◦ = ∅, then K is generating: (Ur(y) ⊆ K, r > 0)

x =

1 2α(y + αx) − 1 2α(y − αx) ∈ K − K

for arbitrary x ∈ X\{0} and 0 < α <

r x.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

For an ordered Banach space X, K and x, y ∈ X one defines

• x ≤ y,

if y − x ∈ K,

• x < y,

if y − x ∈ K\{0},

• x ≪ y,

if y − x ∈ K ◦.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Definition

For an ordered Banach space X, K and x, y ∈ X one defines

• x ≤ y,

if y − x ∈ K,

• x < y,

if y − x ∈ K\{0},

• x ≪ y,

if y − x ∈ K ◦. Clearly x ≥ 0, x > 0 and x ≫ 0 are equivalent to x ∈ K, x ∈ K\{0} and x ∈ K ◦, respectively. The relation ≤ is a partial order on X and x ≪ y ⇒ x < y ⇒ x ≤ y. Moreover, ≤ is compatible with addition, scalar multiplication and convergence.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Example: X = Rn, K = (R+ ∪ {0})n

K is a generating order cone and for x, y ∈ Rn

• x ≤ y,

if xi ≤ yi for all i = 1, . . . , n,

• x < y,

if x ≤ y and xi0 < yi0 for some i0,

• x ≪ y,

if xi < yi for all i = 1, . . . , n.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Example: X = Rn, K = (R+ ∪ {0})n

K is a generating order cone and for x, y ∈ Rn

• x ≤ y,

if xi ≤ yi for all i = 1, . . . , n,

• x < y,

if x ≤ y and xi0 < yi0 for some i0,

• x ≪ y,

if xi < yi for all i = 1, . . . , n. In case n = 1, the relations < and ≪ are equivalent and ≤ corresponds to the common order relation on R.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Example: X = Rn, K = (R+ ∪ {0})n

K is a generating order cone and for x, y ∈ Rn

• x ≤ y,

if xi ≤ yi for all i = 1, . . . , n,

• x < y,

if x ≤ y and xi0 < yi0 for some i0,

• x ≪ y,

if xi < yi for all i = 1, . . . , n. In case n = 1, the relations < and ≪ are equivalent and ≤ corresponds to the common order relation on R.

Example

K = (R+ ∪ {0}) × {0}n−1 is an order cone for X = Rn with K ◦ = ∅. Clearly, K is not total.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Ordered Banach spaces

Example

Consider the (real) Banach space X = C(M) of all continuous, R-valued functions on a compact topological space M equipped with the supremum norm ·∞. K = { f ∈ C(M) : f (x) ≥ 0 for all x ∈ M } is an order cone and for f , g ∈ X

• f ≤ g,

if f (x) ≤ g(x) for all x ∈ M,

• f < g,

if f ≤ g and f (x0) < g(x0) for some x0 ∈ M,

• f ≪ g,

if f (x) < g(x) for all x ∈ M.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Positive operators

Definition

Let X, K be an ordered Banach space. An operator T ∈ B(X) is called

• positive, if T(K\{0}) ⊆ K,
• strictly positive, if T(K\{0}) ⊆ K\{0},
• strongly positive, if T(K\{0}) ⊆ K ◦.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Positive operators

Definition

Let X, K be an ordered Banach space. An operator T ∈ B(X) is called

• positive, if T(K\{0}) ⊆ K,
• strictly positive, if T(K\{0}) ⊆ K\{0},
• strongly positive, if T(K\{0}) ⊆ K ◦.

strongly positive ⇒ strictly positive ⇒ positive

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Positive operators

Let us continue with the previous examples:

Example: X = Rn, K = (R+ ∪ {0})n

Operators A ∈ B(X) can be considered as matrices A ∈ Rn×n.

• A positive: A is a non-negative matrix, i.e. aij ≥ 0.
• A strictly positive: A is a non-negative matrix with at least one

non-zero entry in every row and column.

• A strongly positive: A is a positive matrix, i.e. aij > 0.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Positive operators

Let us continue with the previous examples:

Example: X = Rn, K = (R+ ∪ {0})n

Operators A ∈ B(X) can be considered as matrices A ∈ Rn×n.

• A positive: A is a non-negative matrix, i.e. aij ≥ 0.
• A strictly positive: A is a non-negative matrix with at least one

non-zero entry in every row and column.

• A strongly positive: A is a positive matrix, i.e. aij > 0.

Applying the Krein-Rutman Theorem to this example yields the Perron-Frobenius Theorem.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Positive operators

Example (M ⊆ Rn compact)

X = C(M), K = { f ∈ C(M) : f (x) ≥ 0 for all x ∈ M }. Consider the integral operator T :      X → X f →

• M

A(·, y)f (y) dy with continuous kernel A : M × M → R. The operator T ∈ B(X) is

• compact. If A(x, y) ≥ 0 for all x, y ∈ M, then T is positive. If

A(x, y) > 0 for x, y ∈ M, then T is strongly positive.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Dual cones, ordered dual spaces

Definition

Let X, K be an ordered Banach space. The subset K ′ := { x′ ∈ X ′ : x′, x ≥ 0 for all x ∈ K } ⊆ X ′ is called the dual order cone of K. For x′ ∈ X ′ one defines

• x′ ≥ 0,

if x′ ∈ K ′

• x′ > 0,

if x′ ∈ K ′ and x′, x0 > 0 for some x0 ∈ K

• x′ is strictly positive, if x′, x > 0 for all x ∈ K\{0}

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Dual cones, ordered dual spaces

Definition

Let X, K be an ordered Banach space. The subset K ′ := { x′ ∈ X ′ : x′, x ≥ 0 for all x ∈ K } ⊆ X ′ is called the dual order cone of K. For x′ ∈ X ′ one defines

• x′ ≥ 0,

if x′ ∈ K ′

• x′ > 0,

if x′ ∈ K ′ and x′, x0 > 0 for some x0 ∈ K

• x′ is strictly positive, if x′, x > 0 for all x ∈ K\{0}

In general, K ′ is not an order cone for X ′ and x′ > 0 is not necessarily equivalent to x′ ∈ K ′\{0}.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Dual cones, ordered dual spaces

• If x′ > 0, then x′, x > 0 for all x ∈ K ◦.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Dual cones, ordered dual spaces

• If x′ > 0, then x′, x > 0 for all x ∈ K ◦.
• For every x ∈ K\{0} there exists x′ ∈ K ′ with x′, x > 0.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Dual cones, ordered dual spaces

• If x′ > 0, then x′, x > 0 for all x ∈ K ◦.
• For every x ∈ K\{0} there exists x′ ∈ K ′ with x′, x > 0.
• If K is total, then the dual cone K ′ is an order cone for X ′.

◮ K ′ is closed, K ′ = ∅, and K ′ = {0} ◮ K ′ + K ′ ⊆ K ′ and αK ′ ⊆ K ′ for α > 0 ◮ K ′ ∩ (−K ′) = {0}

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Dual cones, ordered dual spaces

• If x′ > 0, then x′, x > 0 for all x ∈ K ◦.
• For every x ∈ K\{0} there exists x′ ∈ K ′ with x′, x > 0.
• If K is total, then the dual cone K ′ is an order cone for X ′.

◮ K ′ is closed, K ′ = ∅, and K ′ = {0} ◮ K ′ + K ′ ⊆ K ′ and αK ′ ⊆ K ′ for α > 0 ◮ K ′ ∩ (−K ′) = {0}

In this case, positivity of T ∈ B(X) implies positivity of T ′ ∈ B(X ′): T ′x′, x = x′, Tx ≥ 0 x′ ∈ K ′, x ∈ K.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of real Banach spaces

Definition

Consider a real Banach space X. Let the (real) product space X × X be equipped with the regular addition and the complex scalar multiplication (a + ib) · (x, y) := (ax − by, ay + bx) for a + ib ∈ C and (x, y) ∈ X × X. The vector space XC := X × X with the above defined operations is called the complexification of X. Moreover, one defines (x, y)C := max

ϕ∈[0,2π]cos ϕ x + sin ϕ y,

(x, y) ∈ XC.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of real Banach spaces

• XC, ·C is a complex Banach space and ·C induces the product

topology.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of real Banach spaces

• XC, ·C is a complex Banach space and ·C induces the product

topology.

• X can be identified with the closed and (real) subspace

X × {0} ⊆ XC by the isometric and R-linear injection ι :

• X

→ XC x → (x, 0) With this identification, XC can be written as (ix = (0, x)) XC = X ∔R iX with dimR X = dimC XC.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of real Banach spaces

• XC, ·C is a complex Banach space and ·C induces the product

topology.

• X can be identified with the closed and (real) subspace

X × {0} ⊆ XC by the isometric and R-linear injection ι :

• X

→ XC x → (x, 0) With this identification, XC can be written as (ix = (0, x)) XC = X ∔R iX with dimR X = dimC XC.

• There exists a bounded C-isomorphism J : (X ′)C → (XC)′, i.e. one

can identify (X ′)C ≃ (XC)′.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of operators

Definition

Let X be a real Banach space and T ∈ B(X). The complexification of T is defined as TC :

• XC

→ XC x + iy → Tx + iTy acting on the complexification XC of X. Moreover, one defines σ(T) := σ(TC) and r(T) := r(TC).

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of operators

Definition

Let X be a real Banach space and T ∈ B(X). The complexification of T is defined as TC :

• XC

→ XC x + iy → Tx + iTy acting on the complexification XC of X. Moreover, one defines σ(T) := σ(TC) and r(T) := r(TC).

• TC ∈ B(XC) with TC = T.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of operators

Definition

Let X be a real Banach space and T ∈ B(X). The complexification of T is defined as TC :

• XC

→ XC x + iy → Tx + iTy acting on the complexification XC of X. Moreover, one defines σ(T) := σ(TC) and r(T) := r(TC).

• TC ∈ B(XC) with TC = T.
• TC is compact, if and only if T is compact.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of operators

• σp(T) = σp(TC) ∩ R and for λ ∈ σp(T),

geomT(λ) = geomTC(λ) and algT(λ) = algTC(λ)

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Complexification of operators

• σp(T) = σp(TC) ∩ R and for λ ∈ σp(T),

geomT(λ) = geomTC(λ) and algT(λ) = algTC(λ)

• The complexification of the adjoint corresponds to the adjoint of the

complexification via the isomorphism J, J ◦ (TC)′ = (T ′)C ◦ J

• n

(XC)′.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman I

Let X, K be a real, ordered Banach space with total order cone K. Let T ∈ B(X) be compact and positive with r(T) > 0. Then

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman I

Let X, K be a real, ordered Banach space with total order cone K. Let T ∈ B(X) be compact and positive with r(T) > 0. Then

• r(T) ∈ σ(T) is an eigenvalue of T and T ′ ∈ B(X ′).

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman I

Let X, K be a real, ordered Banach space with total order cone K. Let T ∈ B(X) be compact and positive with r(T) > 0. Then

• r(T) ∈ σ(T) is an eigenvalue of T and T ′ ∈ B(X ′).
• There exist positive eigenvectors x0 ∈ K\{0} and x′

0 ∈ K ′\{0} of T

and T ′ respectively, i.e. Tx0 = r(T)x0 > 0 and T ′x′

0 = r(T)x′ 0 > 0.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman I

Let X, K be a real, ordered Banach space with total order cone K. Let T ∈ B(X) be compact and positive with r(T) > 0. Then

• r(T) ∈ σ(T) is an eigenvalue of T and T ′ ∈ B(X ′).
• There exist positive eigenvectors x0 ∈ K\{0} and x′

0 ∈ K ′\{0} of T

and T ′ respectively, i.e. Tx0 = r(T)x0 > 0 and T ′x′

0 = r(T)x′ 0 > 0.

• K is total, so X, K and X ′, K ′ are real, ordered Banach spaces

and both T and T ′ are compact and positive.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Idea of proof: Krein-Rutman I

Consider the complexification S := TC ∈ B(XC) of T, then S and S′ ≃ (T ′)C are compact and every λ ∈ σ(S)\{0} is isolated and an eigenvalue of S and S′. There exists λ ∈ σ(S) with |λ| = r(S).

• Case 1: r(S) ∈ σ(S) is an eigenvalue of S.
• Case 2: λn > 0 for some λ ∈ σ(S) with |λ| = r(S) and n ∈ N.
• Case 3: There is no λ ∈ σ(S) with |λ| = r(S) such that λn > 0 for

some n ∈ N.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman II

Let X, K be a real, ordered Banach space with K ◦ = ∅. Let T ∈ B(X) be compact and strongly positive. Then

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman II

Let X, K be a real, ordered Banach space with K ◦ = ∅. Let T ∈ B(X) be compact and strongly positive. Then

• r(T) > 0 is an algebraically simple eigenvalue of T, i.e.

geomT(r(T)) = algT(r(T)) = 1, with a corresponding eigenvector x0 ∈ K ◦.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman II

Let X, K be a real, ordered Banach space with K ◦ = ∅. Let T ∈ B(X) be compact and strongly positive. Then

• r(T) > 0 is an algebraically simple eigenvalue of T, i.e.

geomT(r(T)) = algT(r(T)) = 1, with a corresponding eigenvector x0 ∈ K ◦.

• ker(λI − T) ∩ K = {0} for λ = r(S).

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman II

Let X, K be a real, ordered Banach space with K ◦ = ∅. Let T ∈ B(X) be compact and strongly positive. Then

• r(T) > 0 is an algebraically simple eigenvalue of T, i.e.

geomT(r(T)) = algT(r(T)) = 1, with a corresponding eigenvector x0 ∈ K ◦.

• ker(λI − T) ∩ K = {0} for λ = r(S).
• |λ| < r(T) for every λ ∈ σ(T)\{r(T)}.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

The Krein-Rutman Theorem

Theorem: Krein-Rutman II

Let X, K be a real, ordered Banach space with K ◦ = ∅. Let T ∈ B(X) be compact and strongly positive. Then

• r(T) > 0 is an algebraically simple eigenvalue of T, i.e.

geomT(r(T)) = algT(r(T)) = 1, with a corresponding eigenvector x0 ∈ K ◦.

• ker(λI − T) ∩ K = {0} for λ = r(S).
• |λ| < r(T) for every λ ∈ σ(T)\{r(T)}.
• The adjoint T ′ ∈ B(X ′) has a strictly positive eigenvector

x′

0 ∈ K ′\{0} corresponding to the algebraically simple eigenvalue

r(T).

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Applications

Corollary

Let X, K be a real, ordered Banach space with K ◦ = ∅ and let T ∈ B(X) be compact and strongly positive.

• Given any v > 0, the inhomogeneous equation

λu − Tu = v, λ ∈ R, has a positive solution u > 0, if and only if λ > r(T). In this case, u is the unique solution.

• If µ, λ ∈ R and u, v > 0 satisfy λu − Tu = µv, then

sgn(µ) = sgn(λ − r(T)).

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

Applications

Corollary

Let X, K be a real, ordered Banach space with K ◦ = ∅ and let T ∈ B(X) be compact and strongly positive. In addition, let P ∈ B(X) be compact.

• If Px ≥ Tx for x ≥ 0, then r(P) ≥ r(T).
• If Px > Tx for all x > 0, then r(P) > r(T).

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem

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