Definable operators on Hilbert spaces The Main Theorem Corollaries - PowerPoint PPT Presentation

Definable operators on Hilbert spaces Isaac Goldbring Continuous Logic Definable operators on Hilbert spaces The Main Theorem Corollaries Isaac Goldbring UCLA ASL North American Annual Meeting Special Session in Model Theory UC Berkeley March 27, 2011

Definable operators on Hilbert spaces Isaac Goldbring Continuous Logic The Main Theorem Continuous Logic Corollaries The Main Theorem Corollaries

Definable Continuous logic in a nutshell operators on Hilbert spaces Isaac Goldbring ◮ Metric structures are bounded complete metric Continuous Logic spaces together with distinguished constants, The Main Theorem functions, and predicates; however, predicates P Corollaries now take values in closed, bounded intervals I P ⊆ R rather than { 0 , 1 } . ◮ The distinguished functions and predicates must also be uniformly continuous. ◮ Metric signatures provide symbols for these distinguished constants, functions and predicates. Moreover, they specify the intervals I P as well as a modulus of uniform continuity for which their interpretations must obey. ◮ For the moment, let’s assume that I P = [ 0 , 1 ] for all predicates P and let us assume that d ≤ 1.

Definable Continuous logic in a nutshell (cont’d) operators on Hilbert spaces Isaac Goldbring Continuous Logic The Main Theorem Corollaries ◮ Atomic formulae are now of the form d ( t 1 , t 2 ) and P ( t 1 , . . . , t n ) , where t 1 , . . . , t n are terms and P is a predicate symbol. ◮ We allow all continuous functions [ 0 , 1 ] n → [ 0 , 1 ] as n -ary connectives. ◮ ∀ x and ∃ x are replaced by sup x and inf x .

Definable Definable predicates operators on Hilbert spaces Isaac Goldbring Continuous Logic ◮ If M is a metric structure and ϕ ( x ) is a formula, The Main Theorem where | x | = n , then the interpretation of ϕ in M is a Corollaries uniformly continuous function ϕ M : M n → [ 0 , 1 ] . ◮ For the purposes of definability, formulae are not expressive enough. Instead, we broaden our perspective to include definable predicates . ◮ If A ⊆ M , then a uniformly continuous function P : M n → [ 0 , 1 ] is definable in M over A if there is a sequence ( ϕ n ( x )) of formulae with parameters from A such that the sequence ( ϕ M n ) converges uniformly to P .

Definable Definable sets and functions operators on Hilbert spaces Isaac Goldbring Continuous Logic ◮ X ⊆ M n is A-definable if and only if X is closed and The Main Theorem the map x �→ d ( x , X ) : M n → [ 0 , 1 ] is an A -definable Corollaries predicate. ◮ f : M n → M is A-definable if and only if the map ( x , y ) �→ d ( f ( x ) , y ) : M n + 1 → [ 0 , 1 ] is an A -definable predicate. ◮ A new complication: Definable sets and functions may now use countably many parameters in their definitions. If the metric structure is separable and the parameterset used in the definition is dense, then this can prove to be troublesome.

Definable Definability takes a backseat operators on Hilbert spaces Isaac Goldbring ◮ There are notions of stability, simplicity, rosiness, Continuous Logic NIP ,... in the metric context. These notions have The Main Theorem been heavily developed with an eye towards Corollaries applications. ◮ However, old-school model theory in the form of definability has not really been pursued. In particular, the question: “Given a metric structure M , what are the sets and functions definable in M?” has not received much attention. The following result appears to be the first result in this direction: Theorem (G.-2010) If U denotes the Urysohn sphere and f : U n → U is definable, then either f is a projection function or has relatively compact image.

Definable Hilbert spaces operators on Hilbert spaces Isaac Goldbring ◮ Throughout, K ∈ { R , C } . Continuous Logic ◮ Recall that an inner product space over K which is The Main Theorem complete with respect to the metric induced by its Corollaries inner product is called a K -Hilbert space. In this talk, H and H ′ denote infinite-dimensional K -Hilbert spaces. ◮ A continuous linear transformation T : H → H ′ is also called a bounded linear transformation. Reason: if one defines � T � := sup {� T ( x ) � : � x � ≤ 1 } , then T is continuous if and only if � T � < ∞ . ◮ We let B ( H ) denote the ( C ∗ -) algebra of bounded operators on H .

Definable Signature for Real Hilbert Spaces operators on Hilbert spaces Isaac Goldbring We work with the following many-sorted metric signature: Continuous Logic ◮ for each n ≥ 1, we have a sort for The Main Theorem Corollaries B n ( H ) := { x ∈ H | � x � ≤ n } . ◮ for each 1 ≤ m ≤ n , we have a function symbol I m , n : B m ( H ) → B n ( H ) for the inclusion mapping. ◮ function symbols + , − : B n ( H ) × B n ( H ) → B 2 n ( H ) ; ◮ function symbols r · : B n ( H ) → B kn ( H ) for all r ∈ R , where k is the unique natural number satisfying k − 1 ≤ | r | < k ; ◮ a predicate symbol �· , ·� : B n ( H ) × B n ( H ) → [ − n 2 , n 2 ] ; ◮ a predicate symbol � · � : B n ( H ) → [ 0 , n ] . The moduli of uniform continuity are the natural ones.

Definable Signature for Complex Hilbert Spaces operators on Hilbert spaces Isaac Goldbring Continuous Logic When working with complex Hilbert spaces, we make the The Main Theorem following changes: Corollaries ◮ We add function symbols i · : B n ( H ) → B n ( H ) for each n ≥ 1, meant to be interpreted as multiplication by i . ◮ Instead of the function symbol �· , ·� : B n ( H ) × B n ( H ) → [ − n 2 , n 2 ] , we have two function symbols Re , Im : B n ( H ) × B n ( H ) → [ − n 2 , n 2 ] , meant to be interpreted as the real and imaginary parts of �· , ·� .

Definable Definable functions operators on Hilbert spaces Definition Isaac Goldbring Let A ⊆ H . We say that a function f : H → H is Continuous Logic A-definable if: The Main Theorem (i) for each n ≥ 1, f ( B n ( H )) is bounded; in this case, we Corollaries let m ( n , f ) ∈ N be the minimal m such that f ( B n ( H )) is contained in B m ( H ) ; (ii) for each n ≥ 1 and each m ≥ m ( n , f ) , the function f n , m : B n ( H ) → B m ( H ) , f n , m ( x ) = f ( x ) is A -definable, that is, the predicate P n , m : B n ( H ) × B m ( H ) → [ 0 , m ] defined by P n , m ( x , y ) = d ( f ( x ) , y ) is A -definable. Lemma The definable bounded operators on H form a subalgebra of B ( H ) .

Definable operators on Hilbert spaces Isaac Goldbring Continuous Logic The Main Theorem Continuous Logic Corollaries The Main Theorem Corollaries

Definable Statement of the Main Theorem operators on Hilbert spaces Isaac Goldbring Continuous Logic From now on, I : H → H denotes the identity operator. The Main Theorem Corollaries Definition An operator K : H → H is compact if K ( B 1 ( H )) has compact closure. (In terms of nonstandard analysis: K is compact if and only if for all finite vectors x ∈ H ∗ , we have K ( x ) is nearstandard.) Theorem (G.-2010) The bounded operator T : H → H is definable if and only if there is λ ∈ K and a compact operator K : H → H such that T = λ I + K.

Definable Statement of the Main Theorem operators on Hilbert spaces Isaac Goldbring Continuous Logic From now on, I : H → H denotes the identity operator. The Main Theorem Corollaries Definition An operator K : H → H is compact if K ( B 1 ( H )) has compact closure. (In terms of nonstandard analysis: K is compact if and only if for all finite vectors x ∈ H ∗ , we have K ( x ) is nearstandard.) Theorem (G.-2010) The bounded operator T : H → H is definable if and only if there is λ ∈ K and a compact operator K : H → H such that T = λ I + K.

Definable Finite-Rank Operators operators on Hilbert spaces ◮ Suppose first that T is a finite-rank operator, that is, Isaac Goldbring T ( H ) is finite-dimensional. Continuous Logic ◮ Let a 1 , . . . , a n be an orthonormal basis for T ( H ) . The Main Theorem Corollaries Then T ( x ) = T 1 ( x ) a 1 + · · · + T n ( x ) a n for some bounded linear functionals T 1 , . . . , T n : H → R . ◮ By the Riesz Representation Theorem, there are b 1 , . . . , b n ∈ H such that T i ( x ) = � x , b i � for all x ∈ H , i = 1 , . . . , n . ◮ Then, for all x , y ∈ H , we have � n n � � � � ( � x , b i � 2 ) − 2 ( � x , b i �� a i , y � ) + � y � 2 d ( T ( x ) , y ) = � i = 1 i = 1 which is a formula in our language. Hence, finite-rank operators are strongly definable.

Definable Compact Operators operators on Hilbert spaces Isaac Goldbring Fact If T : H → H is compact, then there is a sequence ( T n ) of Continuous Logic finite-rank operators such that � T − T n � → 0 as n → ∞ . The Main Theorem Corollaries ◮ Now suppose that T : H → H is a compact operator. Fix a sequence ( T n ) of finite-rank operators such that � T − T n � → 0. ◮ Fix n ≥ 1 and ǫ > 0 and choose k such that � T − T k � < ǫ n . Then for x ∈ B n ( H ) and y ∈ B m ( H ) , where m ≥ m ( n , T ) , we have | d ( T ( x ) , y ) − d ( T k ( x ) , y ) | ≤ � T ( x ) − T k ( x ) � < ǫ. ◮ Since d ( T k ( x ) , y ) is given by a formula, this shows that T is definable. ◮ Thus, any operator of the form λ I + T is definable.