Krdytkyu

    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

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 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)inmathcal{B}(mathcal{H})^d;text{and}; 

    (V_1,cdots,V_d)in mathcal{B}(mathcal{H})^d ;text{such that}\

    displaystylesum_{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} 

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 \

  $displaystylesum_{j=1}^dU_j(T_j-lambda_jI)=I$ and

  $displaystylesum_{j=1}^d(T_j-lambda_jI)V_j=I$.

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.