q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution Deformed commutations of operators and their relations with braided algebras and convolutions Anna Kula 2nd Najman Conference, Dubrovnik 2009 Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution We start with... aa ∗ = a ∗ a normal element ab = commuting elements ba Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution We start with... and complicate things aa ∗ = qa ∗ a q -normal element ab = qba q -commuting elements Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality Definition (general) Let q > 0 and let A be a unital involutive algebra. An element a ∈ A is called q -normal if aa ∗ = qa ∗ a . ◮ such relations often appear in quantum groups ◮ for q = 1 this reduces to the standard notion of normality Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators Definition [S. ˆ Ota, 1998] A densely defined, closed operator M in a Hilbert space H is called q -normal if MM ∗ = qM ∗ M , i.e. MM ∗ f = qM ∗ Mf for all f ∈ D ( MM ∗ ) = D ( M ∗ M ). Remark The classes of q -subnormal, q -hyponormal, q -quasinormal are also studied [for details see ˆ Ota’2002, ˆ Ota’2003, ˆ Ota,Szafraniec’2004, ˆ Ota,Szafraniec’2007]. Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators Basic example: bilateral weighted shift S b : ℓ 2 ( Z ) ⊇ D ( S b ) ∋ { a n } n ∈ Z �→ { w n a n +1 } n ∈ Z ∈ ℓ 2 ( Z ) , where | w n | = | w 0 | q − n 2 , n ∈ N and | w n | 2 | a n +1 | 2 < + ∞} . � D ( S b ) = {{ a n } n ∈ Z ∈ ℓ 2 ( Z ) : n ∈ Z Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators (A choice of) properties [S.ˆ Ota, F.H.Szafraniec] If q � = 1 and M is q -normal, M � = 0, then: Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators (A choice of) properties [S.ˆ Ota, F.H.Szafraniec] If q � = 1 and M is q -normal, M � = 0, then: 1. M – not bounded , not self-adjoint, Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators (A choice of) properties [S.ˆ Ota, F.H.Szafraniec] If q � = 1 and M is q -normal, M � = 0, then: 1. M – not bounded , not self-adjoint, 2. M n is q n 2 -normal, Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators (A choice of) properties [S.ˆ Ota, F.H.Szafraniec] If q � = 1 and M is q -normal, M � = 0, then: 1. M – not bounded , not self-adjoint, 2. M n is q n 2 -normal, 3. qM is unitarily equivalent to M , Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators (A choice of) properties [S.ˆ Ota, F.H.Szafraniec] If q � = 1 and M is q -normal, M � = 0, then: 1. M – not bounded , not self-adjoint, 2. M n is q n 2 -normal, 3. qM is unitarily equivalent to M , 4. 0 ∈ σ ( M ) and L 2 ( σ ( M )) = ∞ , in particular σ ( S b ) = C , Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normal operators (A choice of) properties [S.ˆ Ota, F.H.Szafraniec] If q � = 1 and M is q -normal, M � = 0, then: 1. M – not bounded , not self-adjoint, 2. M n is q n 2 -normal, 3. qM is unitarily equivalent to M , 4. 0 ∈ σ ( M ) and L 2 ( σ ( M )) = ∞ , in particular σ ( S b ) = C , 5. the behaviour depends heavily on whether q < 1 or q > 1, ex. ◮ 0 < q < 1 ⇒ σ r ( M ) = ∅ , ◮ q > 1 ⇒ σ r ( M ) � = ∅ . Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality and q -positivity Observation normal and let f ∈ D ∞ ( M , M ∗ ). Then Let M be n α j � M ∗ ( i + j ) f � 2 ≥ 0 � α i ¯ i , j =0 for any finite seqence of (complex or real) scalars α 0 , . . . , α n . Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality and q -positivity Observation normal and let f ∈ D ∞ ( M , M ∗ ). Then Let M be n α j � M ∗ ( i + j ) f � 2 ≥ 0 � α i ¯ i , j =0 for any finite seqence of (complex or real) scalars α 0 , . . . , α n . Definition A sequence { µ n } n is called positive definite ( PD) if for every finite seqence of scalars α 1 , . . . , α n the following inequality holds n � α i ¯ α j µ i + j ≥ 0 . ( q PD) i , j =0 Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality and q -positivity Observation Let M be q -normal and let f ∈ D ∞ ( M , M ∗ ). Then n α j � M ∗ ( i + j ) f � 2 ≥ 0 � α i ¯ i , j =0 for any finite seqence of (complex or real) scalars α 0 , . . . , α n . Definition A sequence { µ n } n is called positive definite ( PD) if for every finite seqence of scalars α 1 , . . . , α n the following inequality holds n � α i ¯ α j µ i + j ≥ 0 . ( q PD) i , j =0 Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality and q -positivity Observation Let M be q -normal and let f ∈ D ∞ ( M , M ∗ ). Then n α j � M ∗ ( i + j ) f � 2 ≥ 0 � q − ij α i ¯ i , j =0 for any finite seqence of (complex or real) scalars α 0 , . . . , α n . Definition A sequence { µ n } n is called positive definite ( PD) if for every finite seqence of scalars α 1 , . . . , α n the following inequality holds n � α i ¯ α j µ i + j ≥ 0 . ( q PD) i , j =0 Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality and q -positivity Observation Let M be q -normal and let f ∈ D ∞ ( M , M ∗ ). Then n α j � M ∗ ( i + j ) f � 2 ≥ 0 � q − ij α i ¯ i , j =0 for any finite seqence of (complex or real) scalars α 0 , . . . , α n . Definition A sequence { µ n } n is called q -positive definite ( q PD) if for every finite seqence of scalars α 1 , . . . , α n the following inequality holds n � α i ¯ α j µ i + j ≥ 0 . ( q PD) i , j =0 Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution q -normality and q -positivity Observation Let M be q -normal and let f ∈ D ∞ ( M , M ∗ ). Then n α j � M ∗ ( i + j ) f � 2 ≥ 0 � q − ij α i ¯ i , j =0 for any finite seqence of (complex or real) scalars α 0 , . . . , α n . Definition A sequence { µ n } n is called q -positive definite ( q PD) if for every finite seqence of scalars α 1 , . . . , α n the following inequality holds n � q − ij α i ¯ α j µ i + j ≥ 0 . ( q PD) i , j =0 Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution Characterizations of q PD Sequences Recall: By Hamburger Theorem, positive definite sequences are the moment sequences of measures on the real line. Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution Characterizations of q PD Sequences Recall: By Hamburger Theorem, positive definite sequences are the moment sequences of measures on the real line. Theorem The sequence { µ n } n is q PD if and only if there exists a Borel measure µ on R such that � n ( n − 1) t n d µ ( t ) , µ n = q n ∈ N . ( q MS) 2 R Anna Kula: Deformed commutations of operators
q PD + sequences q -normal operators q -commutativity ( p , q )-Commutation ( p , q )-Convolution Characterizations of q PD Sequences Recall: By Hamburger Theorem, positive definite sequences are the moment sequences of measures on the real line. Theorem The sequence { µ n } n is q PD if and only if there exists a Borel measure µ on R such that � n ( n − 1) t n d µ ( t ) , µ n = q n ∈ N . ( q MS) 2 R Definition We call { µ n } n ∈ N a q -moment sequence if there exists a Borel measure µ on (some subset of) R such that ( q MS) holds. Anna Kula: Deformed commutations of operators
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