Matching Tag: operator-theory
Foci conjecture. If cl ( W ( T ) ) \operatorname{cl}(W(T)) cl ( W ( T )) { 0 , 1 , − 1 } \{0,1,-1\} { 0 , 1 , − 1 }
Let A α 2 A^2_\alpha A α 2 D \mathbb{D} D α > − 1 \alpha>-1 α > − 1 M ⊆ A α 2 M\subseteq A^2_\alpha M ⊆ A α 2 S f ( z ) = z f ( z ) S f(z)=z f(z) S f ( z ) = z f ( z )
For every separable Hilbert space E \mathcal E E B ∈ H 1 ∞ ( L ( E ) ) B\in H^\infty_1(L(\mathcal E)) B ∈ H 1 ∞ ( L ( E )) α > − 1 \alpha>-1 α > − 1
Given a bounded vertex-weight family μ = ( μ v i , j ) ( i , j ) ∈ N × N \mu=(\mu_{v_{i,j}})_{(i,j)\in\mathbb{N}\times\mathbb{N}} μ = ( μ v i , j ) ( i , j ) ∈ N × N B μ B_{\mu} B μ
For every integer n ≥ 1 n\ge 1 n ≥ 1 ρ > 1 \rho>1 ρ > 1 z 1 , … , z n z_1,\ldots,z_n z 1 , … , z n ∣ z j ∣ ≤ ρ |z_j|\le \rho ∣ z j ∣ ≤ ρ 1 ≤ j ≤ n 1\le j\le n 1 ≤ j ≤ n
Fix 0 < α , r < ∞ 0<\alpha,r<\infty 0 < α , r < ∞ 0 < p , q ≤ ∞ 0<p,q\leq\infty 0 < p , q ≤ ∞ f ∈ C 1 ( [ − 1 , 1 ] ) f\in C^1([-1,1]) f ∈ C 1 ([ − 1 , 1 ]) f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 ∣ f ′ ( t ) ∣ ≲ ∣ t ∣ α |f'(t)|\lesssim |t|^\alpha ∣ f ′ ( t ) ∣ ≲ ∣ t ∣ α
Fix α > 1 \alpha>1 α > 1 G n ∼ G ( n , α / n ) G_n\sim G(n,\alpha/n) G n ∼ G ( n , α / n ) B n B_n B n
Given Schur-class functions b 1 , b 2 b_1,b_2 b 1 , b 2 D \mathbb{D} D φ : D → D \varphi:\mathbb{D}\to\mathbb{D} φ : D → D
Let A \mathcal A A H H H π : A → B ( H ) \pi:\mathcal A\to\mathcal B(H) π : A → B ( H ) α : A → A \alpha:\mathcal A\to\mathcal A α : A → A
Does there exist an absolute constant C > 0 C>0 C > 0 f ( s ) = ∑ n = 1 N a n n − s f(s)=\sum_{n=1}^{N}a_n n^{-s} f ( s ) = ∑ n = 1 N a n n − s
Given two square nonnegative integer matrices A ∈ N m × m A\in\mathbb{N}^{m\times m} A ∈ N m × m B ∈ N n × n B\in\mathbb{N}^{n\times n} B ∈ N n × n
Let Ω ⊂ Q p d \Omega\subset\mathbb{Q}_p^d Ω ⊂ Q p d D Ω α D_{\Omega}^{\alpha} D Ω α
For each 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ α > − 1 \alpha>-1 α > − 1 g g g J g J_g J g I g I_g I g
Let H 2 ( D ) H^2(\mathbb D) H 2 ( D ) C : H 2 ( D ) → H 2 ( D ) C:H^2(\mathbb D)\to H^2(\mathbb D) C : H 2 ( D ) → H 2 ( D )
Let D = { z ∈ C : ∣ z ∣ < 1 } \mathbb{D}=\{z\in\mathbb{C}:|z|<1\} D = { z ∈ C : ∣ z ∣ < 1 } A ( D ) A(\mathbb{D}) A ( D ) A 2 ( D ) A^2(\mathbb{D}) A 2 ( D ) h ∈ A ( D ) h\in A(\mathbb{D}) h ∈ A ( D )
For every n ∈ N n\in\mathbb{N} n ∈ N P ( z , w ) = ∑ ( j , k ) ∈ F A j , k z j w k ∈ M n ( C [ z ± 1 , w ± 1 ] ) P(z,w)=\sum_{(j,k)\in F}A_{j,k}z^jw^k\in M_n\!\left(\mathbb{C}[z^{\pm1},w^{\pm1}]\right) P ( z , w ) = ∑ ( j , k ) ∈ F A j , k z j w k ∈ M n ( C [ z ± 1 , w ± 1 ] )
For every n , m ≥ 1 n,m\ge 1 n , m ≥ 1 A ∈ M n ( C ) A\in M_n(\mathbb{C}) A ∈ M n ( C ) P ( z ) = ∑ k = 0 d P k z k P(z)=\sum_{k=0}^d P_k z^k P ( z ) = ∑ k = 0 d P k z k P k ∈ M m ( C ) P_k\in M_m(\mathbb{C}) P k ∈ M m ( C )
For every integer n ≥ 3 n\ge 3 n ≥ 3 n n n N N N γ \gamma γ 0 ≤ γ < 2 n n − 1 0\le \gamma<\frac{2n}{n-1} 0 ≤ γ < n − 1 2 n T n # N \mathbb{T}^n\# N T n # N
Let H \mathcal{H} H U , V ∈ B ( H ) U,V\in\mathcal{B}(\mathcal{H}) U , V ∈ B ( H ) ( U − 1 ) ( V − 1 ) (U-\mathbf{1})(V-\mathbf{1}) ( U − 1 ) ( V − 1 ) ( V − 1 ) ( U − 1 ) (V-\mathbf{1})(U-\mathbf{1}) ( V − 1 ) ( U − 1 )
Does there exist an infinite-dimensional Banach space X X X T ∈ L ( X , X ) T\in\mathcal{L}(X,X) T ∈ L ( X , X ) T ∈ L ( X , X ) T\in\mathcal{L}(X,X) T ∈ L ( X , X )
For every integer n ≥ 1 n\ge 1 n ≥ 1 A ∈ C n × n A\in\mathbb{C}^{n\times n} A ∈ C n × n p ∈ C [ z ] p\in\mathbb{C}[z] p ∈ C [ z ] ∥ p ( A ) ∥ 2 ≤ 2 sup z ∈ W ( A ) ∣ p ( z ) ∣ \|p(A)\|_2\le 2\sup_{z\in W(A)}|p(z)| ∥ p ( A ) ∥ 2 ≤ 2 sup z ∈ W ( A ) ∣ p ( z ) ∣
For Hermitian matrices A 1 , A 2 ∈ M n ( C ) A_{1},A_{2}\in M_{n}(\mathbb{C}) A 1 , A 2 ∈ M n ( C ) p ( z 0 , z 1 , z 2 ) = det ( z 0 I + z 1 A 1 + z 2 A 2 ) p(z_{0},z_{1},z_{2})=\det\!\left(z_{0}I+z_{1}A_{1}+z_{2}A_{2}\right) p ( z 0 , z 1 , z 2 ) = det ( z 0 I + z 1 A 1 + z 2 A 2 )
For each n ≥ 1 n\geq 1 n ≥ 1 S n S_n S n n n n X X X T ∈ L ( X ) T\in\mathcal{L}(X) T ∈ L ( X )
For every finitely generated polynomial submodule M ⊂ H 2 ( D 2 ) M\subset H^2(\mathbb D^2) M ⊂ H 2 ( D 2 ) Σ 0 ( M ) ≥ Σ 1 ( M ) ≥ Σ 2 ( M ) ≥ ⋯ \Sigma_0(M)\geq \Sigma_1(M)\geq \Sigma_2(M)\geq\cdots Σ 0 ( M ) ≥ Σ 1 ( M ) ≥ Σ 2 ( M ) ≥ ⋯
Direct-summand conjecture. If
