We give a new proof of continuous Welch bounds obtained by M. Krishna [arXiv:2109.09296]. Our proof is motivated from the proof of Welch [IEEE Transactions on Information Theory, 1974].

We solve the Andô dilation problem for linear maps on vector space asked by Krishna and Johnson in [Oper. Matrices, 2022]. More precisely, we show that any commuting linear maps on vector space can be dilated to commuting injective linear maps.

Let M be an indefinite inner product module over a *-ring of characteristic 2. We show that every self-adjoint operator on M admits Halmos, Egervary and Sz.-Nagy dilations.

A document by K. Mahesh Krishna. Click on the document to view its contents.

We prove the continuous non-Archimedean (resp. p-adic) Banach space and Hilbert space versions of non-Archimedean (resp. p-adic) Welch bounds proved by M. Krishna. We formulate continuous non-Archimedean and p-adic functional Zauner conjectures.

Let K be a non-Archimedean (complete) valued field satisfying n j=1 λ 2 j = max 1≤j≤n |λ j | 2 , ∀λ j ∈ K, 1 ≤ j ≤ n, ∀n ∈ N. For d ∈ N, let K d be the standard d-dimensional non-Archimedean Hilbert space. Let m ∈ N and Sym m (K d) be the non-Archimedean Hilbert space of symmetric m-tensors. We prove the following result. If {τ j } n j=1 is a collection in K d satisfying τ j , τ j = 1 for all 1 ≤ j ≤ n and the operator Sym m (K d) x → n j=1 x, τ ⊗m j τ ⊗m j ∈ Sym m (K d) is diagonalizable, then max 1≤j,k≤n,j =k {|n|, ||τ j , τ k | 2m } ≥ |n| 2 d+m−1 m. (1) We call Inequality (1) as the non-Archimedean version of Welch bounds obtained by Welch [IEEE Transactions on Information Theory, 1974 ]. We formulate non-Archimedean Zauner conjecture.

We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results.

Let p be a prime. For d ∈ N, let Q d p be the standard d-dimensional p-adic Hilbert space. Let m ∈ N and Sym m (Q d p) be the p-adic Hilbert space of symmetric m-tensors. We prove the following result. Let {τ j } n j=1 be a collection in Q d p satisfying (i) τ j , τ j = 1 for all 1 ≤ j ≤ n and (ii) there exists b ∈ Q p satisfying [ n j=1 x, τ j τ j = bx for all x ∈ Q d p. Then max 1≤j,k≤n,j =k {|n|, ||τ j , τ k | 2m } ≥ |n| 2 d+m−1 m. (1) We call Inequality (1) as the p-adic version of Welch bounds obtained by Welch [IEEE Transactions on Information Theory, 1974 ]. Inequality (1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate p-adic Zauner conjecture.

We study C*-algebraic versions of following conjectures/theorems: (1) Bieberbach conjecture (de Branges theorem) (2) Robertson conjecture (3) Lebedev-Milin conjecture (4) Zalcman conjecture (5) Krzyz conjecture (6) Corona conjecture (Carleson theorem). We prove that the C*-algebraic Bieberbach Conjecture for invertible coefficients is true for second degree C*-algebraic polynomials.