The Krein-Rutman Theorem Borbala Mercedes Gerhat Vienna University - PowerPoint PPT Presentation

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 ∈ R n × n , i.e. a ij > 0, the following hold true:

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem The Perron-Frobenius Theorem For a positive matrix A ∈ R n × n , i.e. a ij > 0, the following hold true: • r ( A ) > 0 and r ( A ) ∈ σ ( A ) is algebraically simple, i.e. geom A ( r ( A )) = alg A ( r ( A )) = 1 .

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem The Perron-Frobenius Theorem For a positive matrix A ∈ R n × n , i.e. a ij > 0, the following hold true: • r ( A ) > 0 and r ( A ) ∈ σ ( A ) is algebraically simple, i.e. geom A ( r ( A )) = alg A ( r ( A )) = 1 . • There exists a positive eigenvector associated to r ( A ), Ax = r ( A ) x with x i > 0 .

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem The Perron-Frobenius Theorem For a positive matrix A ∈ R n × n , i.e. a ij > 0, the following hold true: • r ( A ) > 0 and r ( A ) ∈ σ ( A ) is algebraically simple, i.e. geom A ( r ( A )) = alg A ( r ( A )) = 1 . • There exists a positive eigenvector associated to r ( A ), Ax = r ( A ) x with x i > 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 ∈ R n × n , i.e. a ij > 0, the following hold true: • r ( A ) > 0 and r ( A ) ∈ σ ( A ) is algebraically simple, i.e. geom A ( r ( A )) = alg A ( r ( A )) = 1 . • There exists a positive eigenvector associated to r ( A ), Ax = r ( A ) x with x i > 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 α K ⊆ 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 α K ⊆ 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 α K ⊆ 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 . ∈ K ◦ : ( U r (0) ⊆ K , r > 0) • 0 ∈ K , but 0 / r ± 2 � x � x ∈ U r (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 ): n � � � span( K ) ∋ a i x i = a i x i − ( − a i ) x i ∈ K − K a i < 0 i =1 a i ≥ 0

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 ): n � � � span( K ) ∋ a i x i = a i x i − ( − a i ) x i ∈ K − K a i < 0 i =1 a i ≥ 0 • If K ◦ � = ∅ , then K is generating: ( U r ( y ) ⊆ K , r > 0) 1 1 x = 2 α ( y + α x ) − 2 α ( y − α x ) ∈ K − K r for arbitrary x ∈ X \{ 0 } and 0 < α < � 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 , y − x ∈ K \{ 0 } , • x < y , if y − x ∈ K ◦ . • x ≪ y , if

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 , y − x ∈ K \{ 0 } , • x < y , if y − x ∈ K ◦ . • x ≪ y , if 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 = R n , K = ( R + ∪ { 0 } ) n K is a generating order cone and for x , y ∈ R n • x ≤ y , x i ≤ y i for all i = 1 , . . . , n , if • x < y , x ≤ y and x i 0 < y i 0 for some i 0 , if • x ≪ y , if x i < y i for all i = 1 , . . . , n .

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem Ordered Banach spaces Example: X = R n , K = ( R + ∪ { 0 } ) n K is a generating order cone and for x , y ∈ R n • x ≤ y , x i ≤ y i for all i = 1 , . . . , n , if • x < y , x ≤ y and x i 0 < y i 0 for some i 0 , if • x ≪ y , if x i < y i 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 = R n , K = ( R + ∪ { 0 } ) n K is a generating order cone and for x , y ∈ R n • x ≤ y , x i ≤ y i for all i = 1 , . . . , n , if • x < y , x ≤ y and x i 0 < y i 0 for some i 0 , if • x ≪ y , if x i < y i 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 = R n 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 ( x 0 ) < g ( x 0 ) for some x 0 ∈ 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 = R n , K = ( R + ∪ { 0 } ) n Operators A ∈ B ( X ) can be considered as matrices A ∈ R n × n . • A positive: A is a non-negative matrix, i.e. a ij ≥ 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. a ij > 0.

Introduction Ordered Banach spaces, positive operators Krein-Rutman Theorem Positive operators Let us continue with the previous examples: Example: X = R n , K = ( R + ∪ { 0 } ) n Operators A ∈ B ( X ) can be considered as matrices A ∈ R n × n . • A positive: A is a non-negative matrix, i.e. a ij ≥ 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. a ij > 0. Applying the Krein-Rutman Theorem to this example yields the Perron-Frobenius Theorem.