The unilateral shift as a Hilbert module over the disc algebra Rapha¨ el Clouˆ atre Indiana University COSy 2013, Fields Institute May 31, 2013 R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 1 / 18
Hilbert modules over function algebras In 1989, Douglas and Paulsen reformulated several interesting operator theoretic problems using the language of module theory. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 2 / 18
Hilbert modules over function algebras In 1989, Douglas and Paulsen reformulated several interesting operator theoretic problems using the language of module theory.This suggested the use of cohomological methods such as extension groups to further the study of problems such as commutant lifting. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 2 / 18
Hilbert modules over function algebras A bounded linear operator T : H → H is said to be polynomially bounded if there exists a constant C > 0 such that for every polynomial ϕ , we have � ϕ ( T ) � ≤ C � ϕ � ∞ where � ϕ � ∞ = sup | ϕ ( z ) | . | z | < 1 R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 3 / 18
Hilbert modules over function algebras A bounded linear operator T : H → H is said to be polynomially bounded if there exists a constant C > 0 such that for every polynomial ϕ , we have � ϕ ( T ) � ≤ C � ϕ � ∞ where � ϕ � ∞ = sup | ϕ ( z ) | . | z | < 1 The map A ( D ) × H → H ( ϕ, h ) �→ ϕ ( T ) h gives rise to a structure of an A ( D )-module on H , and we say that ( H , T ) is a Hilbert module . R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 3 / 18
Extension groups Given two Hilbert modules ( H 1 , T 1 ) and ( H 2 , T 2 ), we can consider the extension group Ext 1 A ( D ) ( T 2 , T 1 ), which consists of equivalence classes of exact sequences 0 → H 1 → K → H 2 → 0 where K is another Hilbert module and each map is a module morphism. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 4 / 18
Extension groups Given two Hilbert modules ( H 1 , T 1 ) and ( H 2 , T 2 ), we can consider the extension group Ext 1 A ( D ) ( T 2 , T 1 ), which consists of equivalence classes of exact sequences 0 → H 1 → K → H 2 → 0 where K is another Hilbert module and each map is a module morphism. Rather than formally defining the equivalence relation and the group operation, we simply use the following characterization. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 4 / 18
Extension groups Given two Hilbert modules ( H 1 , T 1 ) and ( H 2 , T 2 ), we can consider the extension group Ext 1 A ( D ) ( T 2 , T 1 ), which consists of equivalence classes of exact sequences 0 → H 1 → K → H 2 → 0 where K is another Hilbert module and each map is a module morphism. Rather than formally defining the equivalence relation and the group operation, we simply use the following characterization. Theorem (Carlson-Clark 1995) Let ( H 1 , T 1 ) and ( H 2 , T 2 ) be Hilbert modules. Then, the group Ext 1 A ( D ) ( T 2 , T 1 ) is isomorphic to A / J , where A is the space of operators X : H 2 → H 1 for which the operator � T 1 � X 0 T 2 is polynomially bounded, and J is the space of operators of the form T 1 L − LT 2 for some bounded operator L : H 2 → H 1 . R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 4 / 18
Projective Hilbert modules An important question in the study of extension groups is that of determining which Hilbert modules ( H 2 , T 2 ) have the property that Ext 1 A ( D ) ( T 2 , T 1 ) = 0 for every Hilbert module ( H 1 , T 1 ). R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 5 / 18
Projective Hilbert modules An important question in the study of extension groups is that of determining which Hilbert modules ( H 2 , T 2 ) have the property that Ext 1 A ( D ) ( T 2 , T 1 ) = 0 for every Hilbert module ( H 1 , T 1 ). Such Hilbert modules are said to be projective . R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 5 / 18
Projective Hilbert modules An important question in the study of extension groups is that of determining which Hilbert modules ( H 2 , T 2 ) have the property that Ext 1 A ( D ) ( T 2 , T 1 ) = 0 for every Hilbert module ( H 1 , T 1 ). Such Hilbert modules are said to be projective . Note that T 2 is projective if and only if Ext 1 A ( D ) ( T 1 , T ∗ 2 ) = 0 for every Hilbert module ( H 1 , T 1 ). R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 5 / 18
Projective Hilbert modules An important question in the study of extension groups is that of determining which Hilbert modules ( H 2 , T 2 ) have the property that Ext 1 A ( D ) ( T 2 , T 1 ) = 0 for every Hilbert module ( H 1 , T 1 ). Such Hilbert modules are said to be projective . Note that T 2 is projective if and only if Ext 1 A ( D ) ( T 1 , T ∗ 2 ) = 0 for every Hilbert module ( H 1 , T 1 ). A characterization of projective Hilbert modules has long been sought. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 5 / 18
Results of Carlson and Clark The unilateral shift operator S E : H 2 ( E ) → H 2 ( E ) is defined as ( S E f )( z ) = zf ( z ) for every f ∈ H 2 ( E ). R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 6 / 18
Results of Carlson and Clark The unilateral shift operator S E : H 2 ( E ) → H 2 ( E ) is defined as ( S E f )( z ) = zf ( z ) for every f ∈ H 2 ( E ). Theorem (Carlson-Clark 1995) Let ( H , T) be a Hilbert module. Then, an operator X : H → E gives rise to an element [ X ] ∈ Ext 1 A ( D ) ( T , S E ) if and only if there exists a constant c > 0 such that ∞ � XT n h � 2 ≤ c � h � 2 � n =0 for every h ∈ H . R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 6 / 18
Results of Carlson and Clark The unilateral shift operator S E : H 2 ( E ) → H 2 ( E ) is defined as ( S E f )( z ) = zf ( z ) for every f ∈ H 2 ( E ). Theorem (Carlson-Clark 1995) Let ( H , T) be a Hilbert module. Then, an operator X : H → E gives rise to an element [ X ] ∈ Ext 1 A ( D ) ( T , S E ) if and only if there exists a constant c > 0 such that ∞ � XT n h � 2 ≤ c � h � 2 � n =0 for every h ∈ H . Moreover, for every [ X ] ∈ Ext 1 A ( D ) ( T , S E ) there exists an operator Y : H → E with the property that [ X ] = [ Y ] . R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 6 / 18
Results of Carlson and Clark The unilateral shift operator S E : H 2 ( E ) → H 2 ( E ) is defined as ( S E f )( z ) = zf ( z ) for every f ∈ H 2 ( E ). Theorem (Carlson-Clark 1995) Let ( H , T) be a Hilbert module. Then, an operator X : H → E gives rise to an element [ X ] ∈ Ext 1 A ( D ) ( T , S E ) if and only if there exists a constant c > 0 such that ∞ � XT n h � 2 ≤ c � h � 2 � n =0 for every h ∈ H . Moreover, for every [ X ] ∈ Ext 1 A ( D ) ( T , S E ) there exists an operator Y : H → E with the property that [ X ] = [ Y ] . We bring the reader’s attention to the fact that the group Ext 1 A ( D ) ( T , S E ) is really of a “scalar” nature: it consists of elements [ X ] where the operator X : H → H 2 ( E ) has range contained in the constant functions E . R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 6 / 18
Projective modules in a smaller category Theorem (Ferguson 1997) Let T ∈ B ( H ) be similar to a contraction. The following statements are equivalent: R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 7 / 18
Projective modules in a smaller category Theorem (Ferguson 1997) Let T ∈ B ( H ) be similar to a contraction. The following statements are equivalent: (i) Ext 1 A ( D ) ( T , S E ) = 0 for some separable Hilbert space E R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 7 / 18
Projective modules in a smaller category Theorem (Ferguson 1997) Let T ∈ B ( H ) be similar to a contraction. The following statements are equivalent: (i) Ext 1 A ( D ) ( T , S E ) = 0 for some separable Hilbert space E (ii) the Hilbert module ( H , T ) is projective in the category of Hilbert modules similar to a contractive one R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 7 / 18
Projective modules in a smaller category Theorem (Ferguson 1997) Let T ∈ B ( H ) be similar to a contraction. The following statements are equivalent: (i) Ext 1 A ( D ) ( T , S E ) = 0 for some separable Hilbert space E (ii) the Hilbert module ( H , T ) is projective in the category of Hilbert modules similar to a contractive one (iii) the operator T is similar to an isometry. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 7 / 18
Only known example Theorem (Carlson-Clark-Foias-Williams 1994) If T ∈ B ( H ) is similar to a unitary operator, then the Hilbert module ( H , T ) is projective. R. Clouˆ atre (Indiana University) The unilateral shift as a Hilbert module COSy 2013 8 / 18
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