string input = @"""abc"" <abc@gmail.com>; ""pqr"" <pqr@gmail.com>;";
var output = String.Join(";", Regex.Matches(input, @"<(.+?)>")
                                    .Cast<Match>()
                                    .Select(m => m.Groups[1].Value));

var regex = new Regex(@"<(.*?)>");
        var input= @"""abc"" <abc@gmail.com>;  ""pqr""  <pqr@gmail.com>";
        var matches = regex.Matches(input);
       var res = string.Join(";", matches.Cast<Match>().Select(x => x.Value.Replace("<","").Replace(">","")).ToArray());

string input = ""abc" <abc@gmail.com>; "pqr" <pqr@gmail.com>;";
matchedValuesConcatenated = string.Join(";",
                                Regex.Matches(input, @"(?<=<)([^>]+)(?=>)")
                                .Cast<Match>()
                                .Select(m => m.Value));

string str = ""abc" <abc@gmail.com>; "pqr" <pqr@gmail.com>;";
        string output = string.Empty;
        while (str != string.Empty)
        {
            output += str.Substring(str.IndexOf("<") + 1, str.IndexOf(">") -1);
            str = str.Substring(str.IndexOf(">") + 2, str.Length - str.IndexOf(">") - 2).Trim();
        }

private string GetEmailId(string input)
{
    int start = input.IndexOf("<");
    int end = input.IndexOf(">");
    string result = input.Substring(start + 1, end - start - 1);
    return result;
}

string id = GetEmailId("'abc' <abc@gmail.com>");