A Banach space X has the 2-summing property if the norm of every linear operator from X to a Hilbert space is equal to the 2-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dime...

In this note we prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX − XB is a Hilbert-Schmidt operator, then f(A)X −Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X −Xf(B)||2 ≤ L ||AX −XB||2, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L(H) i...

Two popular model reduction methods, the proper orthogonal decomposition (POD), and balanced truncation, are applied together with Galerkin projection to the twodimensional Burgers’ equation. This scalar equation is chosen because it has a nonlinearity that is similar to the NavierStokes equation, but it can be accurately simulated using far fewer states. However, the number of states required ...

We prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX −XB is a Hilbert-Schmidt operator, then f A X −Xf B is also a Hilbert-Schmidt operator and ‖f A X −Xf B ‖2 ≤ L‖AX −XB‖2 for f belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L H is such that SX − XT belo...

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert-Schmidt kernels are...

We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a very special case. Given n matrices Ai, i = 1, . . . , n, of the same size, let Z1 and Z2 be the block matrices Z1 := (AjA ∗ i ) n i,j=1 and Z2 := (A ∗ jAi) n i,j=1. Then the conjectured inequality is (||Z1||1 − TrZ1) + (||Z2||1 − TrZ2) ≤ ∑ i̸=j ||Ai||2||Aj ||2 ...

Let A be a m × m complex matrix with zero trace. Then there are m ×m matrices B and C such that A = [B,C] and ‖B‖‖C‖2 ≤ (logm + O(1))‖A‖2 where ‖D‖ is the norm of D as an operator on `2 and ‖D‖2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely there is a zero trace m × m matrix A such that whenever A = [B,C], ‖B‖‖C‖2 ≥ | logm−O(1)|‖A‖2 for some abso...

Quantum cloning machines for equatorial qubits are studied. For 1 to 2 phase-covariant quantum cloning machine, using Hilbert-Schmidt norm and Bures fidelity, we show that our transformation can achieve the bound of the fidelity. Networks consisting of quantum gates are presented to realize the quantum cloning transformations. The copied equatorial qubits are shown to be separable by using Pere...

In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and sh...

‎Some improvements of Young inequality and its reverse for positive‎ ‎numbers with Kantorovich constant $K(t‎, ‎2)=frac{(1+t)^2}{4t}$‎ ‎are given‎. ‎Using these inequalities some operator inequalities and‎ ‎Hilbert-Schmidt norm versions for matrices are proved‎. ‎In‎ ‎particular‎, ‎it is shown that if $a‎, ‎b$ are positive numbers and‎ ‎$0 leqslant nu leqslant 1,$ then for all integers $ kgeqsl...