Add a bracket in mathematical equation

\documentclass[12pt,a4paper]{report}
\usepackage{amssymb,amsmath,amsthm,latexsym,mathrsfs,amsfonts,dsfont}
\usepackage{mathtools, nccmath}

\begin{document}
Let $\sigma_H(\mathbf{T})$ denote the Harte spectrum of $\mathbf{T}$
i.e.
$$\lambda=(\lambda_1,\cdots,\lambda_d)\notin \sigma_H(\mathbf{T})$$
$$\Updownarrow$$
$$\exists (U_1,\cdots,U_d)\in \mathcal{B}(\mathcal{H})^d\;\text{and}\; (V_1,\cdots,V_d)\in \mathcal{B}(\mathcal{H})^d \;\text{such that}\;$$
$$\sum_{j=1}^dU_j(T_j-\lambda_jI)=I\;\text{and}\;\sum_{j=1}^d(T_j- \lambda_jI)V_j=I.$$

\end{document}

I get I want to get • Your usage of double dollar signs is odd. You should use the LaTeX equivalents, or better something like gather. – Johannes_B Dec 16 '18 at 10:00
• Why not use a bmatrix env, one column – daleif Dec 16 '18 at 10:34
• @daleif Unfortunalety I don't know how to use it. I hope that you provide me an answer. – Schüler Dec 16 '18 at 10:37
• It is the same thing as geza and joule is doing just simpler, the various matrix envs should be covered in any decent latex introduction. But come to thing of it, the bedt solution is gezas, as it is better semantically – daleif Dec 16 '18 at 10:51

Try

\documentclass[12pt,a4paper]{report}
\usepackage{amssymb,amsmath,amsthm,latexsym,mathrsfs,amsfonts,dsfont}
\usepackage{mathtools, nccmath}

\begin{document}
Let $\sigma_H(\mathbf{T})$ denote the Harte spectrum of $\mathbf{T}$  i.e.
\begin{gather*}
\left(\lambda=(\lambda_1,\cdots,\lambda_d)\notin \sigma_H(\mathbf{T})\right)\\
\Updownarrow\\
\left[
\begin{gathered}
\exists (U_1,\cdots,U_d)\in \mathcal{B}(\mathcal{H})^d\;\text{and}\;
(V_1,\cdots,V_d)\in \mathcal{B}(\mathcal{H})^d \;\text{such that}\;\\
\sum_{j=1}^dU_j(T_j-\lambda_jI)=I\;\text{and}\;\sum_{j=1}^d(T_j-
\lambda_jI)V_j=I.
\end{gathered}\right]
\end{gather*}
\end{document} I wouldn't use parentheses around the top line, it's easy to remove them anyway.

Before going to the proposed solution, some remarks:

1. Never, ever, use $$in LaTeX. 2. Never, ever, use consecutive display math environments. 3. Don't load latexsym: its job is already done by amssymb. 4. amssymb already loads amsfonts. 5. Dots between commas should be low; using \dots will generally give you the right ones (see https://tex.stackexchange.com/a/122497/4427) A stylistic remark: using both \mathcal and \mathscr (from mathrsfs) can confuse readers. In the code below I only load amsmath, the other packages are not necessary for this solution. \documentclass[12pt,a4paper]{report} \usepackage{amsmath} \begin{document} Let \sigma_H(\mathbf{T}) denote the Harte spectrum of \mathbf{T} i.e. \begin{gather*} \bigl(\lambda=(\lambda_1,\dots,\lambda_d)\notin \sigma_H(\mathbf{T})\bigr)\\ \Updownarrow\\ \left[ \begin{tabular}{c} \exists (U_1,\dots,U_d)\in \mathcal{B}(\mathcal{H})^d and (V_1,\dots,V_d)\in \mathcal{B}(\mathcal{H})^d such that \\ \displaystyle\sum_{j=1}^dU_j(T_j-\lambda_jI)=I and \displaystyle\sum_{j=1}^d(T_j-\lambda_jI)V_j=I. \end{tabular} \right] \end{gather*} \end{document} Differences from the good answer by GézàLaTeX: 1. the top line is surrounded by slightly bigger parentheses (but, as said, I think they should be removed); 2. for the bottom block I think it's better using tabular that frees from juggling with \text. • Thank you very much for your great answer. So in my codes of my original document I keep only \usepackage{amssymb,amsmath,amsthm,dsfont,mathrsfs} \usepackage{mathtools, nccmath} Do you agree with me? – Schüler Dec 16 '18 at 11:57 • @Schüler I find the font provided by dsfont ugly as hell, but that's personal preference. Of course if you use it you never use \mathbb. – egreg Dec 16 '18 at 12:03 • So should I delete dsfont? What about \usepackage{mathtools, nccmath}? Is it good? I'm beginner in Latex and thank you for your great help. – Schüler Dec 16 '18 at 12:05 • @Schüler If you don't use \mathds, then the package is redundant; mathtools offers facilities for typesetting math; nccmath offers others, which I find generally of little use. You can easily discover what packages of this kind you're using or not: remove them and compile again, if they are actually used in the document you'll receive errors. As a general advice: don't copy your friend's template. – egreg Dec 16 '18 at 12:10 • @Schüler - The mathtools package loads the amsmath package automatically. – Mico Dec 16 '18 at 13:18 I recommend using  instead of$$ $$. \documentclass[12pt,a4paper]{report} \usepackage{amssymb,amsmath,amsthm,latexsym,mathrsfs,amsfonts,dsfont} \usepackage{mathtools, nccmath} \begin{document} Let \sigma_H(\mathbf{T}) denote the Harte spectrum of \mathbf{T} i.e.$$\Big(\lambda=(\lambda_1,\cdots,\lambda_d)\notin \sigma_H(\mathbf{T})\Big) \Updownarrow 
\left[
\begin{array}{c}
\exists (U_1,\cdots,U_d)\in\mathcal{B}(\mathcal{H})^d\;\text{and}\;
(V_1,\cdots,V_d)\in \mathcal{B}(\mathcal{H})^d \;\text{such that}\\
\displaystyle\sum_{j=1}^dU_j(T_j-\lambda_jI)=I\;\text{and}\;\sum_{j=1}^d(T_j-
\lambda_jI)V_j=I.
\end{array}
\right]
$$\end{document} • You recommend, but you don't show? How about using an amsmath environment? ;-) – Johannes_B Dec 16 '18 at 10:05 • @Johannes_B I want to show that$$ $$is also correct. I don't use  because I'm afraid the OP will think that it is impossible to do the same with$$ $$. Anyway, thanks for your feedback! – user156344 Dec 16 '18 at 10:11 • @JouleV Why  is better that$$ $$?Thanks. – Schüler Dec 16 '18 at 10:23 • @Schüler Why is $…$ preferable to$$? – Johannes_B Dec 16 '18 at 10:52