Subnormal weighted shifts on directed trees whose n th powers have - PowerPoint PPT Presentation

Problem Solution Proof Subnormal weighted shifts on directed trees whose n th powers have trivial domain Zenon Jabło´ nski Instytut Matematyki Uniwersytet Jagiello´ nski joint work with P . Budzy´ nski, I. B. Jung and J. Stochel OTOA 2016 19.12.2016 - Bangalore Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees Let T = ( V , E ) be a directed tree. Let ℓ 2 ( V ) be the space of all square summable function on V with a scalar products � f , g ∈ ℓ 2 ( V ) . � f , g � = f ( u ) g ( u ) , u ∈ V For u ∈ V , let us define e u ∈ ℓ 2 ( V ) by � 1 if u = v , e u ( v ) = 0 if u � = v . { e u } u ∈ V is an orthonormal basis in ℓ 2 ( V ) . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees Let T = ( V , E ) be a directed tree. Let ℓ 2 ( V ) be the space of all square summable function on V with a scalar products � f , g ∈ ℓ 2 ( V ) . � f , g � = f ( u ) g ( u ) , u ∈ V For u ∈ V , let us define e u ∈ ℓ 2 ( V ) by � 1 if u = v , e u ( v ) = 0 if u � = v . { e u } u ∈ V is an orthonormal basis in ℓ 2 ( V ) . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees Let T = ( V , E ) be a directed tree. Let ℓ 2 ( V ) be the space of all square summable function on V with a scalar products � f , g ∈ ℓ 2 ( V ) . � f , g � = f ( u ) g ( u ) , u ∈ V For u ∈ V , let us define e u ∈ ℓ 2 ( V ) by � 1 if u = v , e u ( v ) = 0 if u � = v . { e u } u ∈ V is an orthonormal basis in ℓ 2 ( V ) . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees Let T = ( V , E ) be a directed tree. Let ℓ 2 ( V ) be the space of all square summable function on V with a scalar products � f , g ∈ ℓ 2 ( V ) . � f , g � = f ( u ) g ( u ) , u ∈ V For u ∈ V , let us define e u ∈ ℓ 2 ( V ) by � 1 if u = v , e u ( v ) = 0 if u � = v . { e u } u ∈ V is an orthonormal basis in ℓ 2 ( V ) . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees For a family λ = { λ v } v ∈ V ◦ ⊆ C let us define an operator S λ in ℓ 2 ( V ) by D ( S λ ) = { f ∈ ℓ 2 ( V ): Λ T f ∈ ℓ 2 ( V ) } , S λ f = Λ T f , f ∈ D ( S λ ) , where Λ T is define on functions f : V → C by � if v ∈ V ◦ , � � λ v · f par ( v ) ( Λ T f )( v ) = 0 if v = root . An operator S λ is called a weighted shift on a directed tree T with weights { λ v } v ∈ V ◦ . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees For a family λ = { λ v } v ∈ V ◦ ⊆ C let us define an operator S λ in ℓ 2 ( V ) by D ( S λ ) = { f ∈ ℓ 2 ( V ): Λ T f ∈ ℓ 2 ( V ) } , S λ f = Λ T f , f ∈ D ( S λ ) , where Λ T is define on functions f : V → C by � if v ∈ V ◦ , � � λ v · f par ( v ) ( Λ T f )( v ) = 0 if v = root . An operator S λ is called a weighted shift on a directed tree T with weights { λ v } v ∈ V ◦ . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof A weighted shifts on a directed trees For a family λ = { λ v } v ∈ V ◦ ⊆ C let us define an operator S λ in ℓ 2 ( V ) by D ( S λ ) = { f ∈ ℓ 2 ( V ): Λ T f ∈ ℓ 2 ( V ) } , S λ f = Λ T f , f ∈ D ( S λ ) , where Λ T is define on functions f : V → C by � if v ∈ V ◦ , � � λ v · f par ( v ) ( Λ T f )( v ) = 0 if v = root . An operator S λ is called a weighted shift on a directed tree T with weights { λ v } v ∈ V ◦ . Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Problems Is it true that for every integer n � 1, there exists a subnormal weighted shift on a directed tree whose n th power is densely defined and the domain of its ( n + 1 ) th power is trivial? A similar problem can be stated for composition operators in L 2 -spaces. Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Problems Is it true that for every integer n � 1, there exists a subnormal weighted shift on a directed tree whose n th power is densely defined and the domain of its ( n + 1 ) th power is trivial? A similar problem can be stated for composition operators in L 2 -spaces. Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Characterization Theorem Let S λ be a w.s. on a countably infinite directed tree T = ( V , E ) with weights λ = { λ v } v ∈ V ◦ . Suppose ∃ { µ v } v ∈ V of Borel probability measures on R + and { ε v } v ∈ V ⊆ R + such that � 1 � | λ v | 2 µ u ( ∆ ) = s d µ v ( s ) + ε u δ 0 ( ∆ ) , ∆ ∈ B ( R + ) , u ∈ V . ∆ v ∈ Chi ( u ) (1) Then the following two assertions hold : (i) if S λ is densely defined, then S λ is subnormal, (ii) if n ∈ N , then S n λ is densely defined if and only if � ∞ 0 s n d µ u ( s ) < ∞ for every u ∈ V such that Chi ( u ) has at least two vertices. Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Characterization Theorem Let S λ be a w.s. on a countably infinite directed tree T = ( V , E ) with weights λ = { λ v } v ∈ V ◦ . Suppose ∃ { µ v } v ∈ V of Borel probability measures on R + and { ε v } v ∈ V ⊆ R + such that � 1 � | λ v | 2 µ u ( ∆ ) = s d µ v ( s ) + ε u δ 0 ( ∆ ) , ∆ ∈ B ( R + ) , u ∈ V . ∆ v ∈ Chi ( u ) (1) Then the following two assertions hold : (i) if S λ is densely defined, then S λ is subnormal, (ii) if n ∈ N , then S n λ is densely defined if and only if � ∞ 0 s n d µ u ( s ) < ∞ for every u ∈ V such that Chi ( u ) has at least two vertices. Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Characterization Theorem Let S λ be a w.s. on a countably infinite directed tree T = ( V , E ) with weights λ = { λ v } v ∈ V ◦ . Suppose ∃ { µ v } v ∈ V of Borel probability measures on R + and { ε v } v ∈ V ⊆ R + such that � 1 � | λ v | 2 µ u ( ∆ ) = s d µ v ( s ) + ε u δ 0 ( ∆ ) , ∆ ∈ B ( R + ) , u ∈ V . ∆ v ∈ Chi ( u ) (1) Then the following two assertions hold : (i) if S λ is densely defined, then S λ is subnormal, (ii) if n ∈ N , then S n λ is densely defined if and only if � ∞ 0 s n d µ u ( s ) < ∞ for every u ∈ V such that Chi ( u ) has at least two vertices. Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Lemma Lemma Suppose µ is a finite Borel measure on R + such that � ∞ � ∞ 0 s n d µ ( s ) < ∞ for some n ∈ N . Then 0 s k d µ ( s ) < ∞ for every k ∈ N such that k � n. Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Characterization Lemma Let S λ be a weighted shift on a directed tree T = ( V , E ) with weights λ = { λ v } v ∈ V ◦ and let n ∈ N . Then the following two conditions are equivalent : (i) D ( S n λ ) = { 0 } , ∈ D ( S n (ii) e u / λ ) for every u ∈ V. Moreover, if there exist a family { µ v } v ∈ V of Borel probability measures on R + and a family { ε v } v ∈ V ⊆ R + which satisfy (1) , then (i) is equivalent to � ∞ 0 s n d µ u ( s ) = ∞ for every u ∈ V. (iii) Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Characterization Lemma Let S λ be a weighted shift on a directed tree T = ( V , E ) with weights λ = { λ v } v ∈ V ◦ and let n ∈ N . Then the following two conditions are equivalent : (i) D ( S n λ ) = { 0 } , ∈ D ( S n (ii) e u / λ ) for every u ∈ V. Moreover, if there exist a family { µ v } v ∈ V of Borel probability measures on R + and a family { ε v } v ∈ V ⊆ R + which satisfy (1) , then (i) is equivalent to � ∞ 0 s n d µ u ( s ) = ∞ for every u ∈ V. (iii) Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain

Problem Solution Proof Characterization Lemma Let S λ be a weighted shift on a directed tree T = ( V , E ) with weights λ = { λ v } v ∈ V ◦ and let n ∈ N . Then the following two conditions are equivalent : (i) D ( S n λ ) = { 0 } , ∈ D ( S n (ii) e u / λ ) for every u ∈ V. Moreover, if there exist a family { µ v } v ∈ V of Borel probability measures on R + and a family { ε v } v ∈ V ⊆ R + which satisfy (1) , then (i) is equivalent to � ∞ 0 s n d µ u ( s ) = ∞ for every u ∈ V. (iii) Zenon Jabło´ nski Subnormal operators whose n th powers have trivial domain