Finite section method Algebras of operator sequences Rich sequences Finite sections of convolution type operators Spectral properties of approximation sequences: Helena Mascarenhas IST, University of Lisbon and CEAFEL 14-18 August Joint work with P. Santos and M. Seidel Helena Mascarenhas Approximation operator sequences
Finite section method Algebras of operator sequences Rich sequences Finite sections of convolution type operators Outline of the Talk Finite section method 1 Stability Algebras of operator sequences 2 Convergence notions Algebras of T -structured sequences Fractal algebras Rich sequences 3 T -structured subsequences Passage from sequences to subsequences Finite sections of convolution type operators 4 Multiplication and convolution operators An algebra A of convolution type operators Approximation sequences to operators in A Helena Mascarenhas Approximation operator sequences
Finite section method Algebras of operator sequences Stability Rich sequences Finite sections of convolution type operators Outline of the Talk Finite section method 1 Stability Algebras of operator sequences 2 Convergence notions Algebras of T -structured sequences Fractal algebras Rich sequences 3 T -structured subsequences Passage from sequences to subsequences Finite sections of convolution type operators 4 Multiplication and convolution operators An algebra A of convolution type operators Approximation sequences to operators in A Helena Mascarenhas Approximation operator sequences
Finite section method Algebras of operator sequences Stability Rich sequences Finite sections of convolution type operators Approximate solution to an operator equation Let A ∈ L ( L p ( R )) , with 1 < p < ∞ , and an approximate sequence A n ∈ L (( L p ( R )) of A , based on a sequence of projections P n ∈ L ( L p ( R )) . A common question is to know whether we can substitute the equation u , b ∈ L p ( R ) Au = b , by the “simpler” ones A n u n = P n b and guarantee that u n are unique and converge to the solution of the initial equation. Stability : The sequence ( A n ) is stable if for n large enough the operators A n are invertible and sup � A − 1 n � < ∞ . Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators Outline of the Talk Finite section method 1 Stability Algebras of operator sequences 2 Convergence notions Algebras of T -structured sequences Fractal algebras Rich sequences 3 T -structured subsequences Passage from sequences to subsequences Finite sections of convolution type operators 4 Multiplication and convolution operators An algebra A of convolution type operators Approximation sequences to operators in A Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators Convergence notions Let P = ( P n ) := χ [ − n , n ] I . P -compact operators K ( L p , P ) := { K ∈ L ( L p ( R )) : � K ( I − P n ) � , � ( I − P n ) K � → 0 as n → ∞ } . L ( L p , P ) := { A ∈ L ( L p ( R )) : AK , KA ∈ K ( L p , P ) , ∀ K ∈ K ( L p , P ) } P -Fredholm operators A ∈ L ( L p , P ) is P -Fredholm if A + K ( L p , P ) is invertible in the quotient algebra L ( L p , P ) / K ( L p , P ) . P -Convergence: A sequence ( A n ) ⊂ L ( L p , P ) is said to converge P - strongly to A ∈ L ( L p ( R )) if � K ( A n − A ) � , � ( A n − A ) K � → 0 as n → ∞ for every P -compact operators K . A sequence ( A n ) ⊂ L ( L p ( R )) is said to converge ∗ -strongly to A ∈ L ( L p ( R )) if n → ∞ A ∗ n = A ∗ s - lim n → ∞ A n = A and s - lim Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators Outline of the Talk Finite section method 1 Stability Algebras of operator sequences 2 Convergence notions Algebras of T -structured sequences Fractal algebras Rich sequences 3 T -structured subsequences Passage from sequences to subsequences Finite sections of convolution type operators 4 Multiplication and convolution operators An algebra A of convolution type operators Approximation sequences to operators in A Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators T -structured sequences The set F defined by F := { ( A n ) : A n ∈ L ( L p , P ) and sup n � A n � < ∞ } is a Banach algebra. Consider the following 3 homomorphisms on L p ( R ) : ( V n u )( x ) := u ( x − n ) , ( Z n u )( x ) := n − 1 / p u ( x / n ) and ( U t u )( x ) := e itx u ( x ) , t ∈ R . Let F T be the set of all T -structured sequences , i.e. all sequences A = ( A n ) ∈ F for which the P -strong limits W ± ( A ) := P - lim W ( A ) := P - lim n → ∞ A n , n → ∞ V ∓ n A n V ± n , exist and the ∗ -strong limits n → ∞ Z − 1 n U t A n U − 1 H t ( A ) := s - lim Z n t Z − 1 � � � � also exist for every t ∈ R , where T := ( V ∓ n ) , n U t , t ∈ R . This set forms a closed subalgebra of F . Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators J T -Fredholm sequences F T contains the ideal J T := { ( K )+( V ± n K − V ∓ )+( U − 1 Z n K + Z − 1 n U t )+( G n ) : t K , K − ∈ K ( X , P ) , K + ∈ K and � G n � → 0 as n → ∞ } and we say that ( A n ) ∈ F T is a J T -Fredholm sequence if ( A n )+ J T is invertible in F T / J T . The following theorem is an adaptation of the Silbermann´s lifting theorem Theorem Let A = ( A n ) ∈ F T . Then A is stable if and only if A is J T -Fredholm and W ( A ) , W + ( A ) , W − ( A ) and H t ( A ) , ∀ t ∈ R , are invertible. Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators Outline of the Talk Finite section method 1 Stability Algebras of operator sequences 2 Convergence notions Algebras of T -structured sequences Fractal algebras Rich sequences 3 T -structured subsequences Passage from sequences to subsequences Finite sections of convolution type operators 4 Multiplication and convolution operators An algebra A of convolution type operators Approximation sequences to operators in A Helena Mascarenhas Approximation operator sequences
Finite section method P -compact, P -Fredholm and P -convergence Algebras of operator sequences T -structured sequences Rich sequences P -compact, P -Fredholm and P -convergence Finite sections of convolution type operators Definition Let B be a Banach subalgebra of F containing G := { ( G n ) : � G n � → 0 as n → ∞ } . B is a fractal algebra if for every strictly increasing sequence h of natural numbers and B h := { ( A h n ) : ( A n ) ∈ B } , there exists a map π h : B h → B / G such that for every A ∈ B , A + G = π h ( A h ) . Theorem [Roch, Silbermann 1996] Let p = 2 . If B is a unital fractal subalgebra of F and A = ( A n ) ∈ B then, A is stable if and only if it possesses a stable subsequence. The limit lim � A n � exists and equals � A + G � . Helena Mascarenhas Approximation operator sequences
Finite section method Algebras of operator sequences T -structured subsequences Rich sequences Passage from sequences to subsequences Finite sections of convolution type operators Outline of the Talk Finite section method 1 Stability Algebras of operator sequences 2 Convergence notions Algebras of T -structured sequences Fractal algebras Rich sequences 3 T -structured subsequences Passage from sequences to subsequences Finite sections of convolution type operators 4 Multiplication and convolution operators An algebra A of convolution type operators Approximation sequences to operators in A Helena Mascarenhas Approximation operator sequences
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