Norm Discretization: Dependencies

Let \(\iota \) be a finite nonempty set and put \(m = \# \iota \).

(a) Let \((\Omega , \mu )\) be a measure space and let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate with \(\int _\Omega a(x) a(x)^* \, d\mu (x) = 1\). Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \) and weights \(w_1, \dots , w_n {\gt} 0\) with

(b) Let \(\Omega \) be a finite set and let \(v_k \in \mathbb {C}^{\iota }\), for \(k \in \Omega \), be vectors with \(\sum _{k \in \Omega } v_k v_k^* = 1\). Then for every \(n \geq m\) there are weights \(s_k \geq 0\), at most \(n\) of them nonzero, with

Let \((\Omega , \mu )\) be a measure space, let \(\iota \) be a finite nonempty set, and let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate with positive definite Gram matrix \(I = \int _\Omega a(x) a(x)^* \, d\mu (x)\). Let \(H\) be a nonzero separable complex Hilbert space of measurable functions on \(\Omega \) on which every point evaluation is continuous, let \(K\) be its reproducing kernel, and assume \(\int _\Omega K(x, x) \, d\mu (x) {\lt} \infty \). Let \(\Lambda {\gt} 0\) satisfy \(\int _\Omega |g|^2 \, d\mu \leq \Lambda \| g\| _H^2\) for every \(g \in H\), and assume that the only \(g \in H\) vanishing almost everywhere is \(g = 0\). Put \(m = \# \iota \) and \(M = \int _\Omega K(x, x) \, d\mu (x) / \Lambda \). Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \) and weights \(w_1, \dots , w_n {\gt} 0\) with

for every \(f \in \operatorname {span}\{ a_k\} \), and with

for every \(g \in H\).

Let \((\Omega , \mu )\) be a probability space, let \(\iota \) be a finite nonempty set with \(m = \# \iota \) elements, and let \(a = (a_k)_{k \in \iota }\) be bounded measurable functions on \(\Omega \) with positive definite Gram matrix \(\int _\Omega a(x) a(x)^* \, d\mu (x)\). Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \), not necessarily distinct, with

for every \(f \in \operatorname {span}\{ a_k\} \) and every \(2 \leq p \leq \infty \).

Let \((\Omega , \mu )\) be a probability space, let \(\iota \) be a finite nonempty set with \(m = \# \iota \) elements, and let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate with \(I = \int _\Omega a(x) a(x)^* \, d\mu (x)\) positive definite. Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \), not necessarily distinct, with

Let \(\iota \) be a finite nonempty index set with \(m = \# \iota \) elements, let \(a = (a_k)_{k \in \iota }\) be bounded functions on a set \(\Omega \) with linearly independent coordinate functions, and let \(\varepsilon {\gt} 0\). Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \), not necessarily distinct, such that

for every \(y \in \Omega \) and every \(f \in \operatorname {span}\{ a_k\} \). If \(\Omega \) is a compact space and the \(a_k\) are continuous, the same holds with \(\varepsilon = 0\).

Let \(\iota \) be a finite set, let \(A \in \mathbb {C}^{\iota \times \iota }\) be positive definite, and let \(\delta {\gt} 0\) with \(A - \delta \cdot 1\) positive definite. The lower potential of \(A\) is \(\Phi (A) = \operatorname {Tr}A^{-1}\). With \(Z = (A - \delta \cdot 1)^{-1}\), the lower verifier at \(v \in \mathbb {C}^{\iota }\) is

Let \(H\) be a complex Hilbert space, let \(T\) be a positive operator on \(H\), and let \(e = (e_k)_{k \in \kappa }\) be a Hilbert basis of \(H\). The trace of \(T\) is

and \(T\) is of finite trace if \(\operatorname {Tr}T {\lt} \infty \).

Let \(H\) be a complex Hilbert space, let \(J\) be a positive injective operator on \(H\) of finite trace, let \(B\) be a strictly positive operator on \(H\), that is positive and invertible, and let \(\zeta {\gt} 0\). The upper potential of \(B\) is \(\Psi _J(B) = \operatorname {Tr}\bigl( J B^{-1} \bigr)\), and with \(X = (B + \zeta \cdot J)^{-1}\) the upper verifier at \(u \in H\) is

Let \(\Omega \) be a set, let \(\iota \) be a finite set with \(m = \# \iota \) elements and let \(a : \Omega \to \mathbb {C}^{\iota }\). For all points \(x_1, \dots , x_N \in \Omega \) and weights \(w_1, \dots , w_N \geq 0\) with \(\sum _k w_k = 1\) there are at most \(m^2 + 1\) points \(y_j \in \Omega \) and weights \(v_j \geq 0\) with \(\sum _j v_j = 1\) and \(\sum _j v_j \, a(y_j) a(y_j)^* = \sum _k w_k \, a(x_k) a(x_k)^*\).

Let \((\Omega , \mu )\) be a measure space, let \(\iota \) be a finite set, let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate, and write \(G(a) = \int _\Omega a(x) a(x)^* \, d\mu (x)\).

(a) For every matrix \(Q \in \mathbb {C}^{\iota \times \iota }\),

(b) For every matrix \(S \in \mathbb {C}^{\iota \times \iota }\),

Let \((\Omega , \mu )\) be a measure space, let \(H\) be a complex Hilbert space, let \(b : \Omega \to H\) be square-integrable, and let \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\).

(a) The operators \(b(x) b(x)^*\) are Bochner integrable, and \(J\) is positive with \(\langle u, J u \rangle = \int _\Omega |\langle u, b(x) \rangle |^2 \, d\mu (x)\) for every \(u \in H\).

(b) Let \(\Lambda \in \mathbb {R}\). Then \(J \preceq \Lambda \cdot 1\) if and only if \(\int _\Omega |\langle u, b(x) \rangle |^2 \, d\mu (x) \leq \Lambda \| u\| ^2\) for every \(u \in H\).

(c) \(J\) is injective if and only if every \(u \in H\) with \(\langle u, b(x) \rangle = 0\) for almost every \(x\) is zero.

Let \((\Omega , \mu )\) be a measure space, let \(H\) be a separable complex Hilbert space, let \(b : \Omega \to H\) be square-integrable, and let \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\).

(a) \(J\) is of finite trace, and \(\operatorname {Tr}J = \int _\Omega \| b(x)\| ^2 \, d\mu (x)\).

(b) For every positive bounded operator \(Q\) on \(H\),

Let \(H\) be a complex Hilbert space with a Hilbert basis \(e = (e_k)_{k \in \kappa }\).

(a) For every \(x \in H\) the family \(\bigl( |\langle e_k, x \rangle |^2 \bigr)_k\) has sum \(\| x\| ^2\).

(b) For every bounded operator \(T\) on \(H\),

and the family on the left is summable if and only if the one on the right is.

(c) For every bounded operator \(T\) on \(H\) and every second Hilbert basis \(e' = (e'_j)_{j \in \kappa '}\) of \(H\),

Let \(H\) be a complex Hilbert space of functions on a set \(\Omega \) on which every point evaluation \(f \mapsto f(x)\) is continuous.

(a) For every \(x \in \Omega \) there is \(k_x \in H\) with \(\langle k_x, f \rangle = f(x)\) for every \(f \in H\). With \(K(x, y) = \langle k_y, k_x \rangle \) one has \(K(x, y) = k_x(y)\), so \(k_x = K(x, \cdot )\) and \(K(x, x) = \| k_x\| ^2\), and \(|\langle f, k_x \rangle | = |f(x)|\) for every \(f \in H\).

(b) Let \((\Omega , \mu )\) be a measure space, let \(H\) be separable, and let every \(f \in H\) be measurable. Then \(x \mapsto k_x\) is strongly measurable, and it is square-integrable if \(\int _\Omega K(x, x) \, d\mu (x) {\lt} \infty \).

Let \(\iota \) be a finite set; all matrices below are in \(\mathbb {C}^{\iota \times \iota }\).

(a) Let \(A\) be Hermitian and \(c\) a real number with \(\lambda _i(A) \leq c\) for every eigenvalue. Then \(A \preceq c \cdot 1\).

(b) Every positive semidefinite \(A\) satisfies \(A \preceq (\operatorname {Tr}A) \cdot 1\).

(c) If \(A\) and \(B\) are positive definite with \(A \preceq B\), then \(B^{-1} \preceq A^{-1}\).

(d) If \(A\) is positive definite and \(A^{-1} \preceq c \cdot 1\) for some \(c {\gt} 0\), then \(c^{-1} \cdot 1 \preceq A\).

Let \(\iota \) be a finite set, let \(A \in \mathbb {C}^{\iota \times \iota }\) be positive definite, let \(0 {\lt} \delta {\lt} \Phi (A)^{-1}\) and \(v \in \mathbb {C}^{\iota }\), and let \(w {\gt} 0\) satisfy \(1/w \leq L_A^{\delta }(v)\). Then \(A - \delta \cdot 1 + w \, v v^*\) is positive definite and

Let \(\iota \) be a finite nonempty set, let \(n \geq 1\), and let \(v_1, \dots , v_n \in \mathbb {C}^{\iota }\) and \(w_1, \dots , w_n \in \mathbb {R}\). Let \(r {\gt} 0\) with \(r \neq 1\), put \(\delta = (1 - r)/n\) and \(c_0 = \delta (n r^2 + 1)/r\), and let \(A_n = c_0 \cdot 1 - n \delta \cdot 1 + \sum _{i=1}^n w_i \, v_i v_i^*\). If \(A_n\) is positive definite and \(\Phi (A_n) \leq r / \delta \), then

Let \(\iota \) be a finite set.

(a) Let \(A \in \mathbb {C}^{\iota \times \iota }\) be positive definite and let \(\delta {\gt} 0\) with \(A - \delta \cdot 1\) positive definite. Then

(b) Let \(N \in \mathbb {C}^{\iota \times \iota }\) be positive definite, \(v \in \mathbb {C}^{\iota }\) and \(w \geq 0\). Then

Let \((\Omega , \mu )\) be a measure space, let \(\iota \) be a finite nonempty set, and let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate with \(\int _\Omega a(x) a(x)^* \, d\mu (x) = 1\). Let \(A \in \mathbb {C}^{\iota \times \iota }\) be positive definite and \(0 {\lt} \delta {\lt} \Phi (A)^{-1}\).

(a) The lower verifier satisfies

(b) Let \(U : \Omega \to \mathbb {R}\) be integrable with \(\int _\Omega U \, d\mu \leq 1/\delta - \Phi (A)\). Then there is an \(x \in \Omega \) with \(U(x) {\lt} L_A^{\delta }\bigl( a(x) \bigr)\).

Let \(m \geq 1\) and \(t {\gt} m\) be real. Then \(\alpha = (t-m) / (2m(t-1))\) lies in \([0,1)\) and

Let \(H\) be a complex Hilbert space.

(a) Every positive operator \(T\) on \(H\) of finite trace satisfies \(T \preceq \operatorname {Tr}T \cdot 1\).

(b) Let \(P\) be positive and of finite trace, let \(Q\) be positive, and let \(e = (e_k)_k\) be a Hilbert basis of \(H\). Then

and the value is positive as soon as \(Q\) is invertible and \(P \neq 0\).

(c) Let \(P \preceq P'\) be positive and of finite trace, and let \(Q\) be positive. Then \(\operatorname {Tr}(P Q) \leq \operatorname {Tr}(P' Q)\).

Let \((\Omega , \mu )\) be a measure space and let \(f, g : \Omega \to \mathbb {R}\) be integrable.

(a) If \(\int _\Omega g \, d\mu {\lt} \int _\Omega f \, d\mu \), then \(g(x) {\lt} f(x)\) for at least one \(x \in \Omega \).

(b) If \(f\) and \(g\) are nonnegative with \(\int _\Omega g \, d\mu \leq \int _\Omega f \, d\mu \) and \(\int _\Omega f \, d\mu {\gt} 0\), then there is an \(x \in \Omega \) with \(g(x) \leq f(x)\) and \(f(x) {\gt} 0\).

Let \(\iota \) be a finite set with \(m = \# \iota \) elements, let \(v \in \mathbb {C}^{\iota }\), let \(A \in \mathbb {C}^{\iota \times \iota }\) be invertible, \(\beta \neq 0\) a scalar and \(\alpha \) a scalar. Then

Let \(H\) be a complex Hilbert space and \(u \in H\), and write \(u u^*\) for the rank-one operator \(z \mapsto \langle u, z \rangle u\); for \(H = \mathbb {C}^{\iota }\) it is the matrix \(u u^*\), and \(\langle u, A u \rangle = u^* A u\).

(a) Let \(A\) be a self-adjoint invertible operator on \(H\) and \(t\) a scalar with \(1 + t \langle u, A^{-1} u \rangle \neq 0\). Then \(A + t \, u u^*\) is invertible and

For \(H = \mathbb {C}^{\iota }\) consequently

(b) Let \(A\) be strictly positive and \(w \geq 0\). Then \(A + w \, u u^*\) is strictly positive. If moreover \(w \operatorname {Re}\langle u, A^{-1} u \rangle {\lt} 1\), then \(A - w \, u u^*\) is strictly positive.

(c) \(u u^* \preceq \| u\| ^2 \cdot 1\).

Let \(A\) be a C\(^*\)-algebra with a star ordering, let \(B \in A\) be positive and invertible, let \(X \in A\) and let \(t {\gt} 0\). If \(B^{-1/2} X B^{-1/2} \preceq t \cdot 1\), then \(t^{-1} \cdot X \preceq B\).

Let \(\iota \) be a finite set; all matrices below are in \(\mathbb {C}^{\iota \times \iota }\).

(a) For positive semidefinite \(P\) and \(Q\) we have \(\operatorname {Tr}(P Q) \geq 0\). If \(P\) is positive definite, \(Q\) is positive semidefinite and \(Q \neq 0\), then \(\operatorname {Tr}(P Q) {\gt} 0\).

(b) Let \(Y\) be positive semidefinite and \(Z\) any matrix. Then

(a) Let \(\iota \) be a finite set and let \(A \in \mathbb {C}^{\iota \times \iota }\) be positive definite. Then \(\bigl( \operatorname {Tr}A^{-1} \bigr)^{-1} \cdot 1 \preceq A\).

(b) If moreover \(0 {\lt} \delta {\lt} (\operatorname {Tr}A^{-1})^{-1}\), then \(A - \delta \cdot 1\) is positive definite.

(c) Let \(H\) be a nonzero complex Hilbert space, let \(J\) be a positive injective operator on \(H\) of finite trace, and let \(B\) be a strictly positive operator on \(H\). Then \(\operatorname {Tr}(J B^{-1}) {\gt} 0\) and

Let \(H\) be a nonzero complex Hilbert space, let \(J\) be a positive injective operator on \(H\) of finite trace, let \(B\) be a strictly positive operator on \(H\), let \(\zeta {\gt} 0\), \(u \in H\), and let \(w {\gt} 0\) satisfy \(U_B^{\zeta }(u) \leq 1/w\). Then \(B + \zeta \cdot J - w \, u u^*\) is strictly positive and

Let \(H\) be a nonzero complex Hilbert space, let \(J\) be a positive injective operator on \(H\) of finite trace, let \(\Lambda {\gt} 0\) with \(J \preceq \Lambda \cdot 1\), let \(n \geq 1\), let \(s \geq 1/n\) with \(\operatorname {Tr}J = \Lambda (n s^2 + 1)\), put \(\zeta = (1 + s)/n\) and \(d_0 = \zeta \operatorname {Tr}J / s\), let \(u_1, \dots , u_n \in H\) and \(w_1, \dots , w_n \in \mathbb {R}\), and let \(B_n = d_0 \cdot 1 + n \zeta \cdot J - \sum _{i=1}^n w_i \, u_i u_i^*\). If \(B_n\) is strictly positive and \(\Psi _J(B_n) \leq s / \zeta \), then

Let \(H\) be a nonzero complex Hilbert space, let \(J\) be a positive injective operator on \(H\) of finite trace, and let \(B\) be a strictly positive operator on \(H\).

(a) For every \(\zeta \geq 0\) the operator \(B + \zeta \cdot J\) is strictly positive and

and for \(\zeta {\gt} 0\) the right-hand side is positive.

(b) Let \(C\) be a strictly positive operator on \(H\), let \(u \in H\), and let \(w \geq 0\) with \(w \operatorname {Re}\langle u, C^{-1} u \rangle {\lt} 1\). Then

Let \((\Omega , \mu )\) be a measure space, let \(H\) be a nonzero separable complex Hilbert space, let \(b : \Omega \to H\) be square-integrable with injective Gram operator \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\), let \(B\) be a strictly positive operator on \(H\), and let \(\zeta {\gt} 0\). Then \(x \mapsto U_B^{\zeta }(b(x))\) is integrable and

Let \(\iota \) be a finite nonempty index set with \(m = \# \iota \) elements, let \(a = (a_k)_{k \in \iota }\) be bounded functions on a set \(\Omega \) whose coordinate functions are linearly independent, and let \(\varepsilon {\gt} 0\). Then there are points \(x_1, \dots , x_N \in \Omega \) and weights \(w_1, \dots , w_N \geq 0\) with \(\sum _k w_k = 1\) such that \(G = \sum _k w_k \, a(x_k) a(x_k)^*\) is positive definite and

Let \((\Omega , \mu )\) be a measure space, let \(\iota \) be a finite nonempty set, and let \(H\) be a nonzero separable complex Hilbert space. Let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate with \(\int _\Omega a(x) a(x)^* \, d\mu (x) = 1\), and let \(b : \Omega \to H\) be square-integrable with injective Gram operator \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\). Let \(A_0 \in \mathbb {C}^{\iota \times \iota }\) be positive definite, let \(B_0\) be a strictly positive operator on \(H\), and let \(\delta , \zeta {\gt} 0\) satisfy the initial gap condition

For points \(x_1, \dots , x_k \in \Omega \) and weights \(w_1, \dots , w_k\) write

Then for every \(k \geq 0\) there are points \(x_1, \dots , x_k \in \Omega \) and weights \(w_1, \dots , w_k {\gt} 0\) such that \(A_k\) is positive definite, \(B_k\) is strictly positive, \(\Phi (A_k) \leq \Phi (A_0)\) and \(\Psi _J(B_k) \leq \Psi _J(B_0)\).

Let \((\Omega , \mu )\) be a measure space, let \(\iota \) be a finite nonempty set, and let \(H\) be a nonzero separable complex Hilbert space. Let \(a : \Omega \to \mathbb {C}^{\iota }\) be square-integrable in every coordinate with \(I = G(a) = \int _\Omega a(x) a(x)^* \, d\mu (x)\) positive definite, let \(b : \Omega \to H\) be square-integrable, and let \(J = G(b) = \int _\Omega b(x) b(x)^* \, d\mu (x)\) be its Gram operator; assume that \(J\) is injective. Let \(\Lambda {\gt} 0\) satisfy \(J \preceq \Lambda \cdot 1\), and put \(m = \# \iota \) and \(M = \int _\Omega \| b(x)\| ^2 \, d\mu (x) / \Lambda \). Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \) and weights \(w_1, \dots , w_n {\gt} 0\) with

in the matrix order and

in the order of operators.

Let \(\iota \) be a finite nonempty index set with \(m = \# \iota \) elements, let \(\Omega \) carry a measurable structure, let \(a = (a_k)_{k \in \iota }\) be bounded measurable functions on it with linearly independent coordinate functions, and let \(\varepsilon {\gt} 0\). Then there is a finitely supported probability measure \(\varrho = \sum _{k=1}^N w_k \delta _{x_k}\) on \(\Omega \) whose Gram matrix \(\int _\Omega a(y) a(y)^* \, d\varrho (y) = \sum _k w_k \, a(x_k) a(x_k)^*\) is positive definite and such that

for every \(y \in \Omega \) and every \(f \in \operatorname {span}\{ a_k\} \). If \(\Omega \) is a compact space whose \(\sigma \)-algebra contains the open sets and the \(a_k\) are continuous, the same holds with \(\varepsilon = 0\). In both cases \(\varrho \) can be taken with at most \(N = m^2 + 1\) atoms.

Let \(\iota \) be a finite nonempty index set with \(m = \# \iota \) elements, let \(a = (a_k)_{k \in \iota }\) be bounded functions on a set \(\Omega \) with linearly independent coordinate functions, and let \(\varepsilon {\gt} 0\). Then for every \(n \geq m\) there are points \(x_1, \dots , x_n \in \Omega \), not necessarily distinct, such that

for every \(y \in \Omega \) and every \(f \in \operatorname {span}\{ a_k\} \). If \(\Omega \) is a compact space and the \(a_k\) are continuous, the same holds with \(\varepsilon = 0\).