Deformed commutations of operators and their relations with braided - - PowerPoint PPT Presentation

q-normal operators qPD+ sequences 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-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

We start with...

aa∗ = a∗a normal element ab = ba commuting elements

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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-normal operators qPD+ sequences 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-normal operators qPD+ sequences 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-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-normal operators

Basic example: bilateral weighted shift

Sb : ℓ2(Z) ⊇ D(Sb) ∋ {an}n∈Z → {wnan+1}n∈Z ∈ ℓ2(Z), where |wn| = |w0|q− n

2 ,

n ∈ N and D(Sb) = {{an}n∈Z ∈ ℓ2(Z) :

• n∈Z

|wn|2|an+1|2 < +∞}.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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-normal operators qPD+ sequences 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-normal operators qPD+ sequences 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. Mn is qn2-normal,

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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. Mn is qn2-normal,
• 3. qM is unitarily equivalent to M,

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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. Mn is qn2-normal,
• 3. qM is unitarily equivalent to M,
• 4. 0 ∈ σ(M) and L2(σ(M)) = ∞, in particular σ(Sb) = C,

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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. Mn is qn2-normal,
• 3. qM is unitarily equivalent to M,
• 4. 0 ∈ σ(M) and L2(σ(M)) = ∞, in particular σ(Sb) = C,
• 5. the behaviour depends heavily on whether q < 1 or q > 1, ex.

◮ 0 < q < 1 ⇒ σr(M) = ∅, ◮ q > 1 ⇒ σr(M) = ∅. Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-normality and q-positivity

Observation

Let M be normal and let f ∈ D∞(M, M∗). Then

n

• i,j=0

αi ¯ αjM∗(i+j)f 2 ≥ 0 for any finite seqence of (complex or real) scalars α0, . . . , αn.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-normality and q-positivity

Observation

Let M be normal and let f ∈ D∞(M, M∗). Then

n

• i,j=0

αi ¯ αjM∗(i+j)f 2 ≥ 0 for any finite seqence of (complex or real) scalars α0, . . . , αn.

Definition

A sequence {µn}n is called positive definite ( PD) if for every finite seqence of scalars α1, . . . , αn the following inequality holds

n

• i,j=0

αi ¯ αjµi+j ≥ 0. (qPD)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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

• i,j=0

αi ¯ αjM∗(i+j)f 2 ≥ 0 for any finite seqence of (complex or real) scalars α0, . . . , αn.

Definition

A sequence {µn}n is called positive definite ( PD) if for every finite seqence of scalars α1, . . . , αn the following inequality holds

n

• i,j=0

αi ¯ αjµi+j ≥ 0. (qPD)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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

• i,j=0

q−ijαi ¯ αjM∗(i+j)f 2 ≥ 0 for any finite seqence of (complex or real) scalars α0, . . . , αn.

Definition

A sequence {µn}n is called positive definite ( PD) if for every finite seqence of scalars α1, . . . , αn the following inequality holds

n

• i,j=0

αi ¯ αjµi+j ≥ 0. (qPD)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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

• i,j=0

q−ijαi ¯ αjM∗(i+j)f 2 ≥ 0 for any finite seqence of (complex or real) scalars α0, . . . , αn.

Definition

A sequence {µn}n is called q-positive definite (qPD) if for every finite seqence of scalars α1, . . . , αn the following inequality holds

n

• i,j=0

αi ¯ αjµi+j ≥ 0. (qPD)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences 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

• i,j=0

q−ijαi ¯ αjM∗(i+j)f 2 ≥ 0 for any finite seqence of (complex or real) scalars α0, . . . , αn.

Definition

A sequence {µn}n is called q-positive definite (qPD) if for every finite seqence of scalars α1, . . . , αn the following inequality holds

n

• i,j=0

q−ijαi ¯ αjµi+j ≥ 0. (qPD)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Characterizations of qPD 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-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Characterizations of qPD Sequences

Recall: By Hamburger Theorem, positive definite sequences are the moment sequences of measures on the real line.

Theorem

The sequence {µn}n is qPD if and only if there exists a Borel measure µ on R such that µn = q

n(n−1) 2

• R

tndµ(t), n ∈ N. (qMS)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Characterizations of qPD Sequences

Recall: By Hamburger Theorem, positive definite sequences are the moment sequences of measures on the real line.

Theorem

The sequence {µn}n is qPD if and only if there exists a Borel measure µ on R such that µn = q

n(n−1) 2

• R

tndµ(t), n ∈ N. (qMS)

Definition

We call {µn}n∈N a q-moment sequence if there exists a Borel measure µ on (some subset of) R such that (qMS) holds.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

qPD+ Sequences

Proposition (cf. Stieltjes Theorem)

For a sequence {µn}n the following are equivalent:

• 1. {µn}n is a q-moment sequence corresponding to the measure

µ supported on [0, +∞), that is µn = q

n(n−1) 2

• [0,+∞)

tndµ(t), n ∈ N,

• 2. the sequences {µn}n∈N and {µn+1}n∈N are both qPD.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

qPD+ Sequences

Proposition (cf. Stieltjes Theorem)

For a sequence {µn}n the following are equivalent:

• 1. {µn}n is a q-moment sequence corresponding to the measure

µ supported on [0, +∞), that is µn = q

n(n−1) 2

• [0,+∞)

tndµ(t), n ∈ N,

• 2. the sequences {µn}n∈N and {µn+1}n∈N are both qPD.

We will call such a sequence qPD+.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

qPD+ sequences and states on algebras

Let us consider a unital ∗-algebra A generated by a q-normal a.

1 Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

qPD+ sequences and states on algebras

Let us consider a unital ∗-algebra A generated by a q-normal a.

Remark: state ⇒ qPD+ sequence

If we are given a state φ on A1, then the sequences {µn}n∈N, defined as µn := φ(ana∗n), is qPD+ (is a q-moment sequence corresponding to a measure supported on [0, +∞)).

1linear functional φ : A → C such that φ(1) = 1, φ(x∗x) ≥ 0 for any x ∈ A Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

qPD+ sequences and states on algebras

Let us consider a unital ∗-algebra A generated by a q-normal a.

Remark: state ⇒ qPD+ sequence

If we are given a state φ on A1, then the sequences {µn}n∈N, defined as µn := φ(ana∗n), is qPD+ (is a q-moment sequence corresponding to a measure supported on [0, +∞)).

Remark: state ⇐ qPD+ sequence

If the sequence {µn}n is qPD+, then functional ϕ on A given by ϕ(aka∗l) = δk,lµk is a state on A.

1linear functional φ : A → C such that φ(1) = 1, φ(x∗x) ≥ 0 for any x ∈ A Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

qPD+ sequences and states on algebras

Let us consider a unital ∗-algebra A generated by a q-normal a.

Remark: state ⇒ qPD+ sequence

If we are given a state φ on A1, then the sequences {µn}n∈N, defined as µn := φ(ana∗n), is qPD+ (is a q-moment sequence corresponding to a measure supported on [0, +∞)).

Remark: state ⇐ qPD+ sequence

If the sequence {µn}n is qPD+, then functional ϕ on A given by ϕ(aka∗l) = δk,lµk is a state on A. to be continued...

1linear functional φ : A → C such that φ(1) = 1, φ(x∗x) ≥ 0 for any x ∈ A Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

For two elements a, b of an algebra A, we say that a q-commutes with b (or a and b q-commute) if ab = qba.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

For two elements a, b of an algebra A, we say that a q-commutes with b (or a and b q-commute) if ab = qba.

Remarks

◮ If a q-commutes with b, then b q−1-commutes with a. ◮ If q = 1 and a, b are q-commuting operators on a

(pre-)Hilbert space H, then dim (H) = ∞.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

For two elements a, b of an algebra A, we say that a q-commutes with b (or a and b q-commute) if ab = qba.

Remarks

◮ If a q-commutes with b, then b q−1-commutes with a. ◮ If q = 1 and a, b are q-commuting operators on a

(pre-)Hilbert space H, then dim (H) = ∞.

◮ If ab = qba, then

(a + b)n =

n

• k=0

n k

• q

bn−kak, n ∈ N. Here, [n]q := 1 − qn 1 − q , [n]q! := [1]q·. . .·[n]q, n k

• q

= [n]q! [k]q![n − k]q!.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

Example 1: diagonal operator and shift

Aen = qnen, Ben = en+1, n ∈ N,

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

Example 1: diagonal operator and shift

Aen = qnen, Ben = en+1, n ∈ N,

◮ D is a pre-Hilbert space with the orthonormal basis {en}n∈N,

then we have ABf = qBAf for f ∈ D.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

Example 1: diagonal operator and shift

Aen = qnen, Ben = en+1, n ∈ N,

◮ D is a pre-Hilbert space with the orthonormal basis {en}n∈N,

then we have ABf = qBAf for f ∈ D.

◮ 0 < q < 1, {en}n∈N o.n.b. of a Hilbert space H, then we have

D(AB) = D(BA) = H and ABf = qBAf for f ∈ H.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

Example 1: diagonal operator and shift

Aen = qnen, Ben = en+1, n ∈ N,

◮ D is a pre-Hilbert space with the orthonormal basis {en}n∈N,

then we have ABf = qBAf for f ∈ D.

◮ 0 < q < 1, {en}n∈N o.n.b. of a Hilbert space H, then we have

D(AB) = D(BA) = H and ABf = qBAf for f ∈ H.

◮ q > 1, {en}n∈N o.n.b. of H, then we have

D(AB) = D(BA) = D(A) and ABf = qBAf for f ∈ D(A).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

Example 2: dilation and q-derivative

Let us define the q-derivative of a function f ∈ C[z] as Dqf (z) = f (z) − f (qz) (1 − q)z and the dilation Mqf (z) = f (qz), then MqDq = qDqMq.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity

Example 2: dilation and q-derivative

Let us define the q-derivative of a function f ∈ C[z] as Dqf (z) = f (z) − f (qz) (1 − q)z and the dilation Mqf (z) = f (qz), then MqDq = qDqMq.

Example 3: q-normal operator

If T is q-normal, then there exists a unique contraction KT such that T ∗ ⊃ √qKTT and ker KT ⊇ ker T ∗ [cf. ˆ Ota’2002]. In this case, TKT = qKTT.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

Bialgebra

A bialgebra is an algebra A equipped with the algebra homomorphisms m : A ⊗ A ∋ (a, b) → ab ∈ A, ∆ : A → A ⊗ A, η : C ∋ z → z1 ∈ A, ε : A → C, that satisfy m ◦ (m ⊗ id) = m ◦ (id ⊗ m), (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆, m ◦ (η ⊗ id) = id = m ◦ (id ⊗ η), (ε ⊗ id) ◦ ∆ = id = (id ⊗ ε) ◦ ∆. Here, the structure on A ⊗ A is given by the multiplication rule (a ⊗ b)(c ⊗ d) = ac ⊗ bd and the unit 1 ⊗ 1.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

The multiplication rule on A ⊗ A can be rewritten as mA⊗A = (mA ⊗ mA) ◦ (id ⊗ σ ⊗ id), where σ : A ⊗ A ∋ a ⊗ b → b ⊗ a ∈ A ⊗ A (flip).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

The multiplication rule on A ⊗ A can be rewritten as mA⊗A = (mA ⊗ mA) ◦ (id ⊗ σ ⊗ id), where σ : A ⊗ A ∋ a ⊗ b → b ⊗ a ∈ A ⊗ A (flip). Indeed, we have (a ⊗ b) ⊗ (c ⊗ d) id⊗σ⊗id − →

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

The multiplication rule on A ⊗ A can be rewritten as mA⊗A = (mA ⊗ mA) ◦ (id ⊗ σ ⊗ id), where σ : A ⊗ A ∋ a ⊗ b → b ⊗ a ∈ A ⊗ A (flip). Indeed, we have (a ⊗ b) ⊗ (c ⊗ d) id⊗σ⊗id − → (a ⊗ c) ⊗ (b ⊗ d)

mA⊗mA

− →

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

The multiplication rule on A ⊗ A can be rewritten as mA⊗A = (mA ⊗ mA) ◦ (id ⊗ σ ⊗ id), where σ : A ⊗ A ∋ a ⊗ b → b ⊗ a ∈ A ⊗ A (flip). Indeed, we have (a ⊗ b) ⊗ (c ⊗ d) id⊗σ⊗id − → (a ⊗ c) ⊗ (b ⊗ d)

mA⊗mA

− → ac ⊗ bd

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

The multiplication rule on A ⊗ A can be rewritten as mA⊗A = (mA ⊗ mA) ◦ (id ⊗ σ ⊗ id), where σ : A ⊗ A ∋ a ⊗ b → b ⊗ a ∈ A ⊗ A (flip). Indeed, we have (a ⊗ b) ⊗ (c ⊗ d) id⊗σ⊗id − → (a ⊗ c) ⊗ (b ⊗ d)

mA⊗mA

− → ac ⊗ bd

Generalization

Let Ψ : A ⊗ A → A ⊗ A be a bijectiv linear mapping (YB), called

• braiding. Then the tensor product A ⊗ A has the algebra

structure with respect to the multiplication mΨ mΨ = (mA ⊗ mA) ◦ (id ⊗ Ψ ⊗ id).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

Braided bialgebra

A braided bialgebra is a bialgebra A with multiplication mA, unit η, comutliplication ∆ and counit ε if

• 1. A is a bialgebra with respect to the structure on A ⊗ A

defined by the multiplication mΨ = (mA ⊗ mA) ◦ (id ⊗ Ψ ⊗ id),

• 2. mA, η, ∆ and ε commute with Ψ in the sense that, for

example, Ψ(m ⊗ id) = (id ⊗ m)(Ψ ⊗ id)(id ⊗ Ψ),

• 3. the following relation holds

∆ ◦ m = (m ⊗ m) ◦ (id ⊗ Ψ ⊗ id) ◦ (∆ ⊗ ∆).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

Let

◮ A = C[x] ◮ Ψ(xk ⊗ xm) = qmnxm ⊗ xk (braiding) ◮ ∆(x) = x ⊗ 1 + 1 ⊗ x (comultiplication) ◮ ε(xn) = δn,0 (counit)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

q-commutativity and braided line

Let

◮ A = C[x] ◮ Ψ(xk ⊗ xm) = qmnxm ⊗ xk (braiding) ◮ ∆(x) = x ⊗ 1 + 1 ⊗ x (comultiplication) ◮ ε(xn) = δn,0 (counit)

Then

◮ Cq[x] = (A, ∆, ǫ, m, 1, Ψ) is a braided bialgebra, called

braided line [T. Koornwinder, S. Majid ’90],

◮ a := 1 ⊗ x, b := x ⊗ 1 are generators of A ⊗ A and they satisfy

ab = qba.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

”Complication”: (p, q)-commuting elements

Let p, q > 0 and consider A – the unital ∗-algebra and a, b ∈ A.

Definition

We say that a and b (double) commute with parameters (p, q) or that they (p, q)-commute if ab = pba and ab∗ = qb∗a.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

”Complication”: (p, q)-commuting elements

Let p, q > 0 and consider A – the unital ∗-algebra and a, b ∈ A.

Definition

We say that a and b (double) commute with parameters (p, q) or that they (p, q)-commute if ab = pba and ab∗ = qb∗a.

◮ a and b (p, q)-commute ⇔ a and b (q, p)-commute ◮ a and b (p, q)-commute ⇒ b and a (p−1, q)-commute ◮ example: Aen = qnen, Ben = en+1

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

”Complication”: (p, q)-commuting elements

Let p, q > 0 and consider A – the unital ∗-algebra and a, b ∈ A.

Definition

We say that a and b (double) commute with parameters (p, q) or that they (p, q)-commute if ab = pba and ab∗ = qb∗a.

◮ a and b (p, q)-commute ⇔ a and b (q, p)-commute ◮ a and b (p, q)-commute ⇒ b and a (p−1, q)-commute ◮ example: Aen = qnen, Ben = en+1 (but on ℓ2(Z)!) are

(q, q)-commuting

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

”Compilation”: q-normal, (p, q)-commuting elements

Let p, q > 0, a, b ∈ A and consider the following relations: aa∗ = qa∗a and bb∗ = qb∗b,

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

”Compilation”: q-normal, (p, q)-commuting elements

Let p, q > 0, a, b ∈ A and consider the following relations: aa∗ = qa∗a and bb∗ = qb∗b, ab = pba and ab∗ = qb∗a.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

”Compilation”: q-normal, (p, q)-commuting elements

Let p, q > 0, a, b ∈ A and consider the following relations: aa∗ = qa∗a and bb∗ = qb∗b, ab = pba and ab∗ = qb∗a.

Observation

If a and b are q-normal, (p, q)-commuting, then a + b is again q-normal: (a + b)(a + b)∗ = q(a + b)∗(a + b).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Example: pre-Hilbert Space

◮ D: (complex) pre-Hilbert space with the inner product ., − ◮ L#(D): algebra of all operators A : D → D for which there

exists A# : D → D such that Af , g = f , A#g for f , g ∈ D.

◮ An operator M ∈ L#(D) is q-normal iff

MM#f = qM#Mf for f ∈ D.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Example: pre-Hilbert Space

◮ D: (complex) pre-Hilbert space with the inner product ., − ◮ L#(D): algebra of all operators A : D → D for which there

exists A# : D → D such that Af , g = f , A#g for f , g ∈ D.

◮ An operator M ∈ L#(D) is q-normal iff

MM#f = qM#Mf for f ∈ D.

Observation

Let {em,n}m,n∈Z be an orthonormal basis in D and let Aem,n := p

n 2 q −m−n 2

em+1,n, Bem,n := p− m

2 q −m−n 2

em,n+1. Then A and B are q-normal (in L#(D)) and satisfy the (p, q)-commutation relations.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Example: Braided ∗-bialgebra

braided ∗-bialgebra: A is involutive, ∗-structure is compatibile with the braided bialgebra structure; all homomorphisms are ∗-morphisms w.r.t. the involution on A ⊗ A is given by (a ⊗ b)∗ = Ψ(b∗ ⊗ a∗).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Example: Braided ∗-bialgebra

braided ∗-bialgebra: A is involutive, ∗-structure is compatibile with the braided bialgebra structure; all homomorphisms are ∗-morphisms w.r.t. the involution on A ⊗ A is given by (a ⊗ b)∗ = Ψ(b∗ ⊗ a∗). Let

◮ A = C[x]/xx∗ − qx∗x (∗-algebra genereted by q-normal x) ◮ Ψ(xkx∗l ⊗ xmx∗n) = pln−kmqkn−lmxmx∗n ⊗ xkx∗l (braiding) ◮ ∆(x) = x ⊗ 1 + 1 ⊗ x (comultiplication) ◮ ε(xnx∗m) = δn,0δm,0 (counit)

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

Example: Braided ∗-bialgebra

braided ∗-bialgebra: A is involutive, ∗-structure is compatibile with the braided bialgebra structure; all homomorphisms are ∗-morphisms w.r.t. the involution on A ⊗ A is given by (a ⊗ b)∗ = Ψ(b∗ ⊗ a∗). Let

◮ A = C[x]/xx∗ − qx∗x (∗-algebra genereted by q-normal x) ◮ Ψ(xkx∗l ⊗ xmx∗n) = pln−kmqkn−lmxmx∗n ⊗ xkx∗l (braiding) ◮ ∆(x) = x ⊗ 1 + 1 ⊗ x (comultiplication) ◮ ε(xnx∗m) = δn,0δm,0 (counit)

Then

◮ Cp,q[x, x∗] = (A, ∆, ǫ, m, 1, Ψ, ∗) is a braided ∗-bialgebra ◮ a := x ⊗ 1, b := 1 ⊗ x ⇒ a, b are q-normal, (p, q)-commuting

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

All this stuff – what for? (qPD+ sequences are back!)

Theorem [AK, E. Ricard]

Let a and b be two q-normal, (p, q)-commuting elements and let {µn}n, {νn}n be the qPD+ sequences. Then the mapping Φ[aka∗lbmb∗n] = δk,lδm,nµkνm, is a state on A, the unital ∗-algebra generated by a and b.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

All this stuff – what for? (qPD+ sequences are back!)

Theorem [AK, E. Ricard]

Let a and b be two q-normal, (p, q)-commuting elements and let {µn}n, {νn}n be the qPD+ sequences. Then the mapping Φ[aka∗lbmb∗n] = δk,lδm,nµkνm, is a state on A, the unital ∗-algebra generated by a and b.

Collorary

The functional Φ is a state on the unital ∗-algebra generated by the q-normal operator a + b, and thus the sequence {Φn}n, given by Φn := Φ[(a + b)n(a∗ + b∗)n], is qPD+.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

All this stuff – what for? (qPD+ sequences are back!)

Conclusion

The mapping ({µn}n, {νn}n) → {Φn}n is an operation on qPD+ sequences (preserves the measures on [0, +∞)).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

All this stuff – what for? (qPD+ sequences are back!)

Conclusion

The mapping ({µn}n, {νn}n) → {Φn}n is an operation on qPD+ sequences (preserves the measures on [0, +∞)). But Φn := Φ[(a + b)n(a∗ + b∗)n] can be calculated explicitely and we have Φn =

n

• k=0

q p k(n−k) n k 2

p

µkνn−k.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-Convolution

Definition

Let {µn}n and {νn}n be two sequences. We shall call the (p, q)-convolution of sequences {µn}n and {νn}n, and denote by {(µ ⋆p,q ν)n}n, the sequence given by the formula (µ ⋆p,q ν)n =

n

• k=0

q p k(n−k) n k 2

p

µkνn−k.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-Convolution

Definition

Let {µn}n and {νn}n be two sequences. We shall call the (p, q)-convolution of sequences {µn}n and {νn}n, and denote by {(µ ⋆p,q ν)n}n, the sequence given by the formula (µ ⋆p,q ν)n =

n

• k=0

q p k(n−k) n k 2

p

µkνn−k.

Properties

For any p, q > 0, the (p, q)-convolution is

• 1. associative and commutative,
• 2. symmetric w.r.t. p and p−1: µ1 ⋆p−1,q µ2 = µ1 ⋆p,q µ2,
• 3. preserves the qPD+,
• 4. does not preserve the positive definite or q-positive definite

sequences.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-Fourier Transform

For p, q > 0 we can define the (p, q)-Fourier transform of a measure µ (with q-moments µ(q)

k ):

Fp(µ)(x) =

• Ep(xt)dµ(t),

Ep(x) =

∞

• k=0

(1 − p)2k(−px)k [k]p![k]p−1! ,

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-Fourier Transform

For p, q > 0 we can define the (p, q)-Fourier transform of a measure µ (with q-moments µ(q)

k ):

Fp(µ)(x) =

• Ep(xt)dµ(t),

Ep(x) =

∞

• k=0

(1 − p)2k(−px)k [k]p![k]p−1! , which can be (formally) rewritten as a series with q-moments (for a fixed q > 0). Fp,q(µ)(x) =

∞

• k=0

(−1)kq−k(k−1)/2(1 − p)2kpkxk [k]p![k]p−1! µ(q)

k .

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-Fourier Transform

For p, q > 0 we can define the (p, q)-Fourier transform of a measure µ (with q-moments µ(q)

k ):

Fp(µ)(x) =

• Ep(xt)dµ(t),

Ep(x) =

∞

• k=0

(1 − p)2k(−px)k [k]p![k]p−1! , which can be (formally) rewritten as a series with q-moments (for a fixed q > 0). Fp,q(µ)(x) =

∞

• k=0

(−1)kq−k(k−1)/2(1 − p)2kpkxk [k]p![k]p−1! µ(q)

k .

Proposition

Fp,q(µ ⋆p,q ν)(x) = Fp,q(µ)(x) · Fp,q(ν)(x).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-convolution as a convolution in a braided ∗-bialgebra

For a braided ∗-bialgebra (A, ∆, ǫ, m, 1, Ψ, ∗), the convolution of two functionals on A is defined µ ⋆ ν := (µ ⊗ ν) ◦ ∆.

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-convolution as a convolution in a braided ∗-bialgebra

For a braided ∗-bialgebra (A, ∆, ǫ, m, 1, Ψ, ∗), the convolution of two functionals on A is defined µ ⋆ ν := (µ ⊗ ν) ◦ ∆. Given two qPD+ sequences {µn}, {νn} and the functionals µ(xmx∗m) := δm,nµm, ν(xmx∗m) = δm,nνm we have (µ ⋆p,q ν)n = (µ ⊗ ν) ◦ ∆(xnx∗n). Thus the (p, q)-convolution is a convolution in Cp,q[x, x∗]:

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

(p, q)-convolution as a convolution in a braided ∗-bialgebra

For a braided ∗-bialgebra (A, ∆, ǫ, m, 1, Ψ, ∗), the convolution of two functionals on A is defined µ ⋆ ν := (µ ⊗ ν) ◦ ∆. Given two qPD+ sequences {µn}, {νn} and the functionals µ(xmx∗m) := δm,nµm, ν(xmx∗m) = δm,nνm we have (µ ⋆p,q ν)n = (µ ⊗ ν) ◦ ∆(xnx∗n). Thus the (p, q)-convolution is a convolution in Cp,q[x, x∗]:

Remark

In case p = q or p = q−1, the positivity of convolution is guaranteed by Theorem by U. Franz, R. Schott (’1996).

Anna Kula: Deformed commutations of operators

q-normal operators qPD+ sequences q-commutativity (p, q)-Commutation (p, q)-Convolution

References

• 1. S. ˆ

Ota, Some classes of q-deformed operators, J. Operator Theory 48 (2002), 151-186.

• 2. T. H. Koornwinder, Special functions and q-commumting

variables, Fields Institute Communications 14 (1997), 127-166.

• 3. A. Kula, E. Ricard, On a convolution for q-normal operators,

Infinite Dimensional Analysis, Quantum Probability and Related Topics 11(4) (2008), 565-588.

Anna Kula: Deformed commutations of operators