Fredholm operators on interpolation spaces Mieczysaw Mastyo Adam - PowerPoint PPT Presentation

Fredholm operators on interpolation spaces Mieczysław Mastyło Adam Mickiewicz University, Poznań Workshop on Banach spaces and Banach lattices ICMAT, September 9–13, 2019 Based on joint works with Irina Asekritova and Natan Kruglyak M. Mastyło (UAM) Fredholm operators on interpolation spaces 1 / 39

Outline 1 Interpolation functors, Fredholm operators 2 The domination property for interpolation functors 3 The uniqueness of inverses on intersection of a Banach couple 4 The uniqueness of inverses on intersection of interpolated Banach spaces 5 Appendix M. Mastyło (UAM) Fredholm operators on interpolation spaces 2 / 39

Interpolation functors, Fredholm operators Interpolation functors, Fredholm operator) Definition A mapping F : � B → B from the category � B of all couples of Banach spaces into the category B of all Banach spaces is said to be an interpolation functor (or method) if, for any couple � X := ( X 0 , X 1 ), the Banach space F ( X 0 , X 1 ) is intermediate with respect to � X (i.e., X 0 ∩ X 1 ⊂ F ( X 0 , X 1 ) ⊂ X 0 + X 1 ), and T : F ( X 0 , X 1 ) → F ( Y 0 , Y 1 ) for all T : ( X 0 , X 1 ) → ( Y 0 , Y 1 ) , where T : ( X 0 , X 1 ) → ( Y 0 , Y 1 ) means that T : X 0 + X 1 → Y 0 + Y 1 is a linear operator such that the restrictions T | X 0 : X 0 → Y 0 , T | X 1 : X 1 → Y 1 are bounded operators. Remark. The space of all operators T : � X → � Y is a Banach space equipped with the norm � � � T � � Y := max � T | X 0 � X 0 → Y 0 , � T | X 1 � X 1 → Y 1 X → � M. Mastyło (UAM) Fredholm operators on interpolation spaces 4 / 39

Interpolation functors, Fredholm operators • The real method. For θ ∈ (0 , 1) and p ∈ [1 , ∞ ], ( X 0 , X 1 ) θ, p is defined as the Banach space of all x ∈ X 0 + X 1 equipped with the norm � � ∞ � 1 / p � p dt t − θ K ( t , x ; � � � x � θ, p = X ) , t 0 where K ( t , x ; � X ) := inf {� x 0 � X 0 + t � x 1 � X 1 ; x = x 0 + x 1 } , t > 0 . M. Mastyło (UAM) Fredholm operators on interpolation spaces 5 / 39

Interpolation functors, Fredholm operators • The complex method. Let S := { z ∈ C ; 0 < Re z < 1 } be an open strip on the plane. For a given θ ∈ (0 , 1) and any couple � X = ( X 0 , X 1 ) we denote by F ( � X ) the Banach space of all bounded continuous functions f : ¯ S → X 0 + X 1 on the closure ¯ S that are analytic on S , and R ∋ t �→ f ( j + it ) ∈ X j , j = 0 , 1 is a bounded continuous function, and equipped with the norm � � � f � F ( � X ) = max sup � f ( it ) � X 0 , sup � f (1 + it ) � X 1 . t ∈ R t ∈ R The (lower) complex interpolation space [ � X ] θ := { f ( θ ); f ∈ F ( � X ) } and is equipped with the norm: � � � x � θ := inf � f � F ( � X ) ; f ( θ ) = x . M. Mastyło (UAM) Fredholm operators on interpolation spaces 6 / 39

Interpolation functors, Fredholm operators • Variants of the complex method. Let B be the class of all Banach spaces over the complex field. A mapping X : B → B is called a pseudolattice lattice (on Z ), if it satisfy the following conditions: (i) for every B ∈ B the space X ( B ) consists of B valued sequences { b n } = { b n } n ∈ Z modelled on Z ; (ii) whenever A is a closed subspace of B it follows that X ( A ) is a closed subspace of X ( B ); (iii) there exists a positive constant C such that, for all A , B ∈ B and all bounded linear operators T : A → B and every sequence { a n } ∈ X ( A ), the sequence { Ta n } ∈ X ( B ) and satisfies the estimate �{ Ta n }� X ( B ) � C � T � A → B �{ a n }� X ( A ) ; (iv) � b m � B � �{ b n }� X ( B ) for each m ∈ Z , all { b n } ∈ X ( B ) and all Banach spaces B . M. Mastyło (UAM) Fredholm operators on interpolation spaces 7 / 39

Interpolation functors, Fredholm operators • For every Banach couple � B = ( B 0 , B 1 ) and every couple of pseudolattices X = ( X 0 , X 1 ): � � B → � B , let J ( � X , � B ) be the Banach space of all B 0 ∩ B 1 valued sequences { b n } n ∈ Z such that { e jn b n } n ∈ Z ∈ X j ( B j ) ( j = 0 , 1), equipped with the norm. � �{ b n }� X 0 ( B 0 ) , �{ e n b n }� X 1 ( B 1 ) � �{ b n }� J ( � B ) = max . X ,� • Following Cwikel–Kalton–Milman–Rochberg (2002), for every s in the annulus A := { z ∈ C ; 1 < | z | < e } , we define the Banach space � X , s to B � consist of all elements of the form b = � n ∈ Z s n b n (convergence in B 0 + B 1 with { b n } ∈ J ( � X , � B ), equipped with the norm � � � s n b n � b � � X , s = inf �{ b n }� J ( � B ) ; b = . X ,� B � n ∈ Z The map � B �→ � X , s is an interpolation method (on � B � B ). M. Mastyło (UAM) Fredholm operators on interpolation spaces 8 / 39

Interpolation functors, Fredholm operators • A couple � X = ( X 0 , X 1 ) of Banach pseudolattices, is said to be translation invariant if for any Banach space B , � � { S k ( { b n } n ∈ Z �� � � X j ( B ) = � { b n } n ∈ Z X j ( B ) , j ∈ { 0 , 1 } � � for all { b n } n ∈ Z ∈ X j ( B ), each k ∈ Z , where S is the left-shift operator defined by S { b n } = { b n +1 } . • � X = ( X 0 , X 1 ) is said to be a rotation invariant Banach couple of pseudolattices whenever the rotation map { b n } n ∈ Z �→ { e in τ b n } n ∈ Z is an isometry of X j ( B ) onto itself for every real τ and every Banach space B . M. Mastyło (UAM) Fredholm operators on interpolation spaces 9 / 39

Interpolation functors, Fredholm operators Definition A bounded linear operator T : X → Y between Banach spaces is said to be semi-Fredholm if T ( X ) is closed in Y and at least one of the spaces ker T , Y / T ( X ) is finite-dimensional. Then the index of T is given by ind( T ) := dim(ker T ) − dim( Y / T ( X )) . If ind( T ) is finite, T is called a Fredholm operator. Properties: (1) If T : X → Y is a Fredholm operator, then the dual operator T ∗ : Y ∗ → X ∗ is also Fredholm and ind( T ∗ ) = − ind( T ) . (2) If T : X → Y and S : Y → Z are Fredholm operators, then ST : X → Z is also a Fredholm operator with ind( ST ) = ind( T ) + ind( S ) . M. Mastyło (UAM) Fredholm operators on interpolation spaces 10 / 39

Interpolation functors, Fredholm operators (3) A strictly singular perturbation of a Fredholm operator remains Fredholm and has the same index, i.e., if T : X → Y is a Fredholm operator and S : X → Y is a strictly singular operator, then T + S is a Fredholm operator and ind( T + S ) = ind( T ) . (4) If X is a Banach space and S : X → X is a strictly singular (in particular a compact) operator, then I X − λ S is a Fredholm operator for every λ with ind ( I X − λ S ) = 0 . (5) Every Fredholm operator T : X → Y between Banach spaces has a pseudoinverse which is also Fredholm operator, i.e., such an operator S : Y → X satisfying : TST = T . In particular this yields that the equation Tx = y has a solution if and only if Sy is a solution of this equation. M. Mastyło (UAM) Fredholm operators on interpolation spaces 11 / 39

Interpolation functors, Fredholm operators Theorem ( Atkinson ) For an operator T : X → Y between Banach spaces the following statements are equivalent : (i) T is Fredholm operator. (ii) There exist compact ( finite rank ) operators K 1 : X → X and K 2 : Y → Y and an operator S : Y → X such that ST = I X − K 1 and TS = I Y − K 2 . Theorem ( Kato ) If T : X → Y is a Fredholm operator between Banach spaces, then for any operator S : X → Y such that � T � < γ ( S ) := inf {� Sx � Y ; dist ( x , ker S ) > 0 } . Then T + S is Fredholm with dim ( ker ( T + S )) � dim ( ker T ) , ind ( T + S ) = ind ( T ) . M. Mastyło (UAM) Fredholm operators on interpolation spaces 12 / 39

Interpolation functors, Fredholm operators Theorem ( I. Ya. Shneiberg, 1974 ) Let T : ( X 0 , X 1 ) → ( Y 0 , Y 1 ) be an operator between Banach couples. Then the following statements are true : (i) If T : [ X 0 , X 1 ] θ ∗ → [ Y 0 , Y 1 ] θ ∗ is invertible for some θ ∗ ∈ (0 , 1) , then there exists ε > 0 such that T : [ X 0 , X 1 ] θ ∗ → [ Y 0 , Y 1 ] θ ∗ is invertible for all θ ∈ ( θ ∗ − ε, θ ∗ + ε ) . (ii) If T : [ X 0 , X 1 ] θ ∗ → [ Y 0 , Y 1 ] θ ∗ is Fredholm for some θ ∗ ∈ (0 , 1) , then there exists ε > 0 such that T : [ X 0 , X 1 ] θ → [ Y 0 , Y 1 ] θ is Fredholm and the index is constant for all θ ∈ ( θ ∗ − ε, θ ∗ + ε ) . M. Mastyło (UAM) Fredholm operators on interpolation spaces 13 / 39

Interpolation functors, Fredholm operators Let Ω ⊂ R n ( n � 2) be a domain above the graph of real-valued Lipschitz function defined in R n − 1 � i.e., Ω = { ( x , φ ( x ) + t ); x ∈ R n − 1 , t > 0 } , where φ : R n − 1 → R is a Lipschitz function � . Question: For which 1 < p < ∞ the Dirichlet problem for the Laplacian: ∆ u = 0 in Ω ( ∗ ) under the conditions M ( u ) ∈ L p ( ∂ Ω) and u | ∂ Ω = f ∈ L p ( ∂ Ω) has a solution? Here, M stands for the nontangential maximal operator given by M ( u )( x ) := sup {| u ( y ) | ; y ∈ Ω , | x − y | < 2 dist( y , ∂ Ω) } , x ∈ ∂ Ω and u | ∂ Ω is defined (in the sense of nontangential convergence to the boundary) by u | ∂ Ω ( x ) := lim u ( y ) , x ∈ ∂ Ω . Ω ∋ y → x | x − y | < 2 dist( y ,∂ Ω) M. Mastyło (UAM) Fredholm operators on interpolation spaces 14 / 39